Keywords: programme de Langlands local p-adique et modulo p, compatibilité local-global, Fontaine-Laffaille, sommes de Jacobi, représentations de Steinberg, famille de représentations localement analytiques, fonction dilogarithme p-adique p-adic and mod p local Langlands program, local-global compatibility, Fontaine-Laffaille, Jacobi sums, Steinberg representation, family of locally analytic representations, p-adique dilogarithm function v vi

Titre: la compatibilité local-global p-adique et modulo p pour GL n (Q p ) Cette thèse est consacrée à deux aspects du programme de Langlands local p-adique et de la compatibilité local-global p-adique.

Dans la première partie, j'étudie la question de savoir comment extraire, d'un certain sous-espace Hecke-isotypique de formes automorphes modulo p, suffisament d'invariants d'une représentation galoisienne. Soient p un nombre premier, n > 2 un entier, et F un corps à multiplication complexe dans lequel p est complètement décomposé. Supposons qu'une représentation galoisienne automorphe continue r : Gal(Q/F ) → GL n (F p ) est triangulaire supérieure, Fontaine-Laffaille et suffisament générique (dans un certain sens) en une place w au-dessus de p. On montre, en admettant un résultat d'élimination de poids de Serre prouvé dans [LLMPQ], que la classe d'isomorphisme de r| Gal(Q p /Fw) est déterminée par l'action de GL n (F w ) sur un espace de formes automorphes modulo p découpé par l'idéal maximal associée à r dans une algèbre de Hecke. En particulier, on montre que la partie sauvagement ramifiée de r| Gal(Q p /Fw) est déterminée par l'action de sommes de Jacobi (vus comme éléments de F p [GL n (F p )]) sur cet espace.

. Soient E une extension finie de Q p suffisamment grande et ρ p : Gal(Q p /Q p ) → GL 3 (E) une représentation p-adique semi-stable telle que la représentation de Weil-Deligne WD(ρ p ) associée a un opérateur de monodromie N de rang 2 et que la filtration de Hodge associée est noncritique. On sait que la filtration de Hodge de ρ p dépend de trois invariants dans E. On construit une famille de représentations localement analytiques Σ min (λ, L 1 , L 2 , L 3 ) qui dépend de trois invariants L 1 , L 2 , L 3 ∈ E et telle que chaque représentation contient la représentation localement algébrique Alg ⊗ Steinberg déterminée par ρ p . Quand ρ p provient, pour un groupe unitaire convenable G /Q , d'une représentation automorphe π de G(A Qp ) avec un niveau fixé U p premier avec p, on montre (sous quelques hypothèses techniques) qu'il existe une unique représentation localement analytique dans la famille ci-dessus qui est une sous-représentation du sous-espace Hecke-isotypique associé dans la cohomologie complétée de niveau U p . On rappelle que [Bre17] a construit une famille de représentations localement analytiques qui dépend de quatre invariants (voir (4) dans [Bre17]) avec une propriété similaire. On donne un critère purement de théorie de représentation: si une représentation Π dans la famille de Breuil se plonge dans un certain sous-espace Hecke-isotypique de la cohomologie complétée, alors elle se plonge nécessairement dans une Σ min (λ, L 1 , L 2 , L 3 ) pour certains choix de L 1 , L 2 , L 3 ∈ E qui sont déterminés explicitement par Π. De plus, certains sousquotients naturels de Σ min (λ, L 1 , L 2 , L 3 ) permettent de construite un complexe de représentations localement analytiques qui réalise l'objet dérivé abstrait Σ(λ, L ) defini dans [Schr11]. Par conséquent, la famille de représentations localement analytiques Σ min (λ, L 1 , L 2 , L 3 ) fournit une relation entre les L -invariants supérieurs étudiés dans [Bre17] et [BD18], et la fonction dilogarithme p-adique qui est apparue dans la construction de Σ(λ, L ) dans [Schr11].

I would like to thank Prof. Florian Herzig for offering me the chance of a postdoctoral position at University of Toronto and for answering many questions from me through the past few years. I am also very honored to invite him to my defense.

I would like to thank Prof. Luc Illusie for allowing me to visit him shortly after I came to Paris. I could not think of contacting my supervisor without his extremely kind suggestions and personal introduction, which probably has already changed my life.

I would like to thank Prof. Chol Park as a kind friend and collaborator since an early stage of my Ph.D. I made many mathematical mistakes at the beginning and he encouraged him a lot throughout our collaboration. I may not be able to dive into certain technical problems for a long time without his understanding. His carefulness during research do save me from making more mistakes.

I would like to thank Prof. Yongquan Hu and Prof. Yiwen Ding for their generous explanation on countless mathematical problems. They also offer me great opportunity to visit Beijing each Summer vocation to attend their seminars and meet the other mathematicians.

I would like to thank Prof. Viet Bao Le Hung and Daniel Le for their suggestions on an early draft of this thesis and especially for their help on the improvement of my understanding on moduli of Fontaine-Laffaille modules and Breuil-Kisin modules. I gradually have more conceptual and geometric understandings around this area thanks to their insights.

I would like to thank Prof. Julien Hauseux for inviting me to Université de Lille twice and for our discussions there.

I would like to thank Prof. James Edward Humphreys for kindly answering my questions via email at an early stage of my Ph.D. His two books on BGG category O and on modular representations of finite groups of Lie type respectively are always very helpful references.

I would like to thank Prof. Frédéric Paulin, Prof. Stephane Nonnenmacher, Madame Marie-Christine Myoupo and Monsieur Nicolas Olleon for kindly helping me finish all the administrative procedure for graduation at Université Paris-sud.

I would like to thank ENS d'Ulm for accepting me as a student via concours sélection internationale. I could not imagine how would I reach so much modern mathematics and many strong mathematicians without coming to ENS. I would like to thank Prof. Claude Viterbo, Prof. Jan Nekovar and Prof. Lie Fu for their tutorial help when I was a student at ENS. I also would like to thank Madame Isabelle vii viii ACKNOWLEDGEMENT Mistral and Monsieur Auguste Filippi for helping me get used to the life at ENS. I would like to thank Shengyuan Zhao for helping me translate the introduction of my thesis into French.

I would like to thank my friends Chen Hu, Kaitong Hu, Censi Li, Jialun Li, Finski Siarhei, Uladzislau Stazhynski, Ruoci Sun, Shengquan Xiang, Junyi Zhang, Hui Zhu who came to ENS at the same time as me. We had a great time living and studying together at ENS. I had the chance to meet many friends at ENS including Yichang Cai, Yawen Chen, Ji Dai, Zhihao Duan, Hao Fu, Songyuan Li, Linyuan Liu, Xiao Ma, Jian Qian, Jiaxin Qiao, Yichen Qin, Jieao Song, Yijun Wan, Hua Wang, Baojun Wu, Minchen Xia, Lizao Ye, Hao Zhang, Haowen Zhang, Yi Zhang, Yizhen Zhao, Peng Zheng, Zechuan Zheng, Deliang Zhong for their company at ENS. I would like to thank the friends I have met at Université Paris-sud and IHES including Yang Cao, Linxiao Chen, Zhangchi Chen, Jingren Chi, Wei-Guo Foo, Ning Guo, Binxiao Liu, Kegang Liu, Jinrui Niu, Jie Ren, Changjian Su, Yisheng Tian, Xiaozong Wang, Songyan Xie, Daxin Xu, Cong Xue, Yeping Zhang, Peng Zhou, Jiandi Zou for their help and company.

I would like to thank many young mathematicians I have met in Paris including Xiaohua Ai, Chuqi Cao, Jiaming Chen, Peiyi Cui, Xianglong Duan, Yanbo Fang, Chenlin Gu, Huajie Li, Xingyu Li, Yongqi Liang, Jie Lin, Chenguang Liu, Chunhui Liu, Shinan Liu, Wille Liu, Qiaochu Ma, Yu Min, Shu Shen, Haoran Wang, Shanwen Wang, Disheng Xu, Ruotao Yang, Hongjie Yu, Xiaoyu Zhang. I think Ph.D students in mathematics have many things in common and it is great that we have been sharing thoughts with each other.

I would like to thank some other Chinese friends in France including Yifan Chen, Yue Feng, Weichen Gao, Enhui Huang, Yichen Liu, Momo, Heshu Wang, Haowu Wang, Min Wang, Xinyang Wang, Zhengjian Wang, Tairan Xu, Fan Yang, Chaoping Zhang as we had a lot of fun together with my girlfriend.

I would like to thank the Séminaire Mathjeune which has offered me a chance to be an organizer. Although I am lazy and not so efficient as an organizer, I do have met many young Ph.D students and Postdocs through this Seminar. I would like to thank all the former organizers as well.

I would like to thank Prof. Yi Ouyang and Prof. Xinan Ma for writing recommendation letters for me during my application for ENS d'Ulm.

I would like to thank Prof. Xiaonan Ma for his generous help for me and my girlfriend, especially for helping us renting a comfortable apartment at Bourg la Reine.

I would like to thank Prof. Yang Chen at SCGY of USTC who still cares about my development after my graduation. I would like to thank Prof. Jihuai Shi and Prof. Guangtian Song for teaching me calculus and linear algebra which is important throughout mathematical research. I would like to thank Prof. Qin Chen, Prof. Simin Li, Prof. Congwen Liu, Prof. Guangbin Ren, Prof. Bin Xu, and many other professors for teaching me mathematics at USTC. I would like to thank Madame Naying Li for helping me overcome many non-academic difficulties at USTC. I was relatively young when I entered USTC as undergraduate student and I am really grateful that she was sufficiently patient to listen to my personal feelings. I would like to thank Yifan Quan, Xinzhou Tu and Wei Shuai for being my roommates for three years at USTC. I would like to thank my amazing friends at USTC who kindly encouraged me and guided me through many challenges.

I would like to thank an anonymous person online who suggested me to apply for ENS d'Ulm. I would like to thank Meiya Wang and Xiaohong Wang who taught me mathematics at primary school. They offered me the first chance to enjoy the pleasure of mathematics. I would like to thank Yuanchun Wang who taught me the importance of rigorous proof during middle school and Xiaodong Zha who gradually convinced me during high school that I could probably become a professional mathematician in the future.

I would like to thank my parents for their full support and especially for allowing me to dive into something I am truly interested in. I am extremely luck because I have nothing to worry about except my thesis problem during Ph.D. I would like to thank my girlfriend Yi Pan for her company throughout my Ph.D. It is always very challenging and in particular time consuming to solve some non-trivial mathematical problems. Therefore it is important that I can share my feelings with her when I am stuck by something annoying. We have overcome many difficulties together and this is the best part of my life in Paris. I also would like to thank her parents for their support.

There are still many other names that has contributed to my life. I could not list them all but just would like to thank them all for their help.

Chapter 1

Introduction générale

1.1 Conjecture de compatibilité locale-globale p-adique Soit p un nombre premier, K une extension finie de Q p et E une autre extension finie de Q p qui est suffisamment grande. On note O K (resp. K , resp. k) l'anneau des entiers (resp. une uniformisante, resp. le corps résiduel) de K; On note O E (resp.

E , resp. k E ) l'anneau des entiers (resp. une uniformisante, resp. le corps résiduel) de E. Le programme de Langlands local p-adique initié par Breuil dans [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) I[END_REF], [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) II[END_REF] et [START_REF] Breuil | Invariant L et série spéciale p-adique[END_REF] vise à associer une représentation linéaire de GL n (K) pLL(ρ) sur un espace de Banach p-adique à une représentation galoisienne p-adique ρ : Gal(K/K) → GL n (E). Cette correspondance de Langlands locale p-adique est connue pour GL 2 (Q p ) et est dû à Colmez dans [START_REF] Colmez | Représentations de GL 2 (Q p ) et ( , Γ)-modules[END_REF]; sa compatibilité avec la cohomologie étale complétée de la tour de courbes modulaires est montrée par Emerton dans [START_REF] Emerton | A local-global compatibility conjecture in the p-adic Langlands programme for GL 2 /Q Pure and Applied Math[END_REF]. L'application pLL reste encore très mystérieuse quand K = Q p ou quand n ≥ 3. On s'attend (c.f. [START_REF] Breuil | The emerging p-adic Langlands programme[END_REF]) à ce que pLL soit compatible avec la reduction modulo E , avec les familles p-adiques , etc. Nous utilisons dans la suite la notation pLL pour désigner l'application (qui est seulement conjecturale pour K = Q p ou n ≥ 3) qui associe à une représentation galoisienne ρ : Gal(K/K) → GL n (k E ) une representation lisse admissible de GL n (K) à coefficients dans k E .

On commence par donner une formulation plus précise de cette compatibilité local-global conjecturale pour pLL, qui est une généralisation naturelle des idées dans [START_REF] Emerton | A local-global compatibility conjecture in the p-adic Langlands programme for GL 2 /Q Pure and Applied Math[END_REF] (sauf que nous ne connaissons plus l'existence de pLL si K = Q p ou si n ≥ 3). À partir de maintenant on fixe un corps F + totalement réel et une extension F quadratique totalement imaginaire de F + . On fixe une place finie v 0 de F + qui divise p, qui est scindée dans F et qui vérifie K ∼ = F + v0 ∼ = F w0 où w 0 est une place finie de F au-dessus de v 0 . On fixe une telle place w 0 de F à partir de maintenant. On fixe un groupe unitaire G sur F + tel que

(i) G ⊗ F + F ∼ = GL n / F ; (ii) G(F + ⊗ Q R) est compact.
On fixe aussi un sous-groupe compact ouvert U v0 ⊂ G(A ∞,v0 F + ) et un O E -module A de type fini muni de sa topologie p-adique. Puis on considère l'espace des formes automorphes p-adiques continues (resp. localement constantes) à valeurs dans A sur G(A ∞ F + ) de un niveau fixé U v0 premier avec v 0 :

S(U v0 , A) := {f : G(F + )\G(A ∞ F + )/U v0 → A, continue } ( resp. S(U v0 , A) := {f : G(F + )\G(A ∞ F + )/U v0 → A, localement constante } ). En particulier si A est un module de p-torsion muni de la topologie discrète, alors les deux espaces définis ci-dessus coïncident. Chaque espace ci-dessus admet une action d'une algèbre de Hecke universelle T sur O E (engendrée librement par des opérateurs de Hecke indexés par les places finies v de F au-dessus d'un ensemble D(U v0 ) de places finies de F + qui sont totalement décomposées dans F , premières avec p, telles que U v := U v0 ∩ G(F + v ) est un sous-groupe compact ouvert maximal de G(F + v )); et il y a aussi une action continue ( resp. lisse) de GL n (K) sur S(U v0 , A) (resp. sur S(U v0 , A)) qui provient de la translation à droite sur G(F + )\G(A ∞ F + )/U v0 . Les actions de GL n (K) sur les deux espaces commutent avec les actions de T. Pour une représentation galoisienne p-adique continue r A : Gal(F /F ) → GL n (A) qui est conjuguée auto-duale et non-ramifiée à toute place de F au-dessus d'un v ∈ D(U v0 ), on peut lui associer un idéal p r A ⊂ T ⊗ O E A; Le sous-espace propre S(U v0 , A)[p r A ] (resp. S(U v0 , A)[p r A ]) défini par cet idéal admet naturellement une action continue (resp. lisse) de GL n (K). On note LL l'application donnée par la correspondance de Langlands locale classique qui envoie une représentation de Weil-Deligne, Frobenius semi-simple de dimension n, vers la représentation lisse irréductible de GL n (K) (c.f. [START_REF] Harris | The geometry and cohomology of some simple Shimura varieties AM-151[END_REF], [START_REF] Henniart | Une preuve simple des conjectures de Langlands pour GL n sur un corps p-adique[END_REF] et [START_REF] Scholze | The Local Langlands Correspondence for GL n over p-adic fields Invent[END_REF]). On pose

r := r O E ⊗ O E E, r := r k E = r O E ⊗ O E k E , ρ := r| Gal(K/K) , ρ := r| Gal(K/K)
et on utilise la notation abrégée

Π(r) := S(U v0 , O E )[p r O E ] ⊗ O E E Π(r) := S(U v0 , k E )[p r k E ] Π sm (r) := S(U v0 , O E )[p r O E ] ⊗ O E E .
On omet le niveau U v0 dans la notation pour simplicité. On définit Π alg (r) comme le sousespace de Π(r) des vecteurs localement algébriques. On remarque que Π alg (r) est naturellement une représentation localement algébrique de GL n (K). On note WD(ρ) (resp. Alg(ρ)) l'application qui associe une représentation de Weil-Deligne (resp. une représentation algébrique de GL n (K) de dimension finie) à une représentation galoisienne p-adique ρ potentiellement semi-stable (resp. l'ensemble des poids de Hodge-Tate de ρ) via la théorie de Fontaine de [START_REF] Fontaine | Représentations p-adiques semi-stables[END_REF] (resp. via la section 5 de [START_REF] Breuil | Vers le socle localement analytique pour GL n I[END_REF]). A ce stade, la compatibilité local-global classique en p (qui a été demontrée dans [START_REF] Barnet-Lamb | Local-global compatibility for = p I[END_REF] et [START_REF] Barnet-Lamb | Local-global compatibility for = p II[END_REF] sous plus d'hypothèses techniques) peut se formuler grossièrement comme suit :

Théorème 1.1.1. Supposons que Π alg (r) = 0.

Alors ρ est potentiellement semi-stable et il existe un entier d > 0 qui ne dépend que de r et de U v0 tel que Π alg (r) ∼ = (LL • WD(ρ) ⊗ E Alg(ρ)) ⊕d .

(1.1.2)

Il est naturel d'imaginer que la compatibilité local-global p-adique conjecturale devrait avoir la forme suivante:

Espoir 1.1.3. Supposons que Π(r) = 0.

Alors il existe un entier d > 0 qui ne dépend que de r et de U v0 tel que Π(r) ∼ = pLL(ρ) ⊕d (1.1.4) et Π(r) ∼ = pLL(ρ) ⊕d .

(1.1.5)

En particulier, l'Espoir 1.1.3 implique :

Conjecture 1.1.6. La structure de Π(r) (resp. de Π(r)) comme représentation de Banach p-adique unitaire admissible (resp. représentation lisse admissible à coefficients dans k E ) de GL n (K) détermine la classe d'isomorphisme de ρ (resp. de ρ) et ne dépend que de cette classe.

On insiste sur le fait que ρ n'est pas nécessairement potentiellement semi-stable dans l'Espoir 1.1.3 et dans la Conjecture 1.1.6. En revanche, comme les applications pLL et pLL sont très mystérieuses quand K = Q p ou quand n ≥ 3, on ne considère que les cas où ρ est potentiellement semi-stable dans le reste de cette introduction de sorte que le Théorème 1.1.1 soit disponible. Il est assez difficile d'étudier directement les représentations de Banach p-adiques unitaires de GL n (K). On a essentiellement deux types d'objets par lesquels il est peut-être plus facile de commencer. Ma thèse est donc divisée en deux parties et dans chaque partie on étudie l'un de ces deux types d'objets. La première partie de ma thèse traite les représentations Π(r) lisses et admissibles à coefficients dans k E (travail en commun avec Chol Park) et la seconde partie traite les représentations Π an (r) localement analytiques définies comme l'ensemble des vecteurs localement analytiques (l'ensemble des vecteurs sur lesquels GL n (K) agit par des fonctions localement analytiques sur le groupe GL n (K) à valeurs vectorielles) dans Π(r).

Compatibilité local-global modulo p pour GL n (Q p )

On commence par Π(r). Plusieures questions naturelles se posent sur Π(r):

(i) Est-ce qu'on a Π(r) = 0? (ii) Quelle est la structure de Π(r)? (iii) Quel est le lien explicite entre Π(r) et ρ = r| Gal(K/K) (on rappelle que K = F w0 )? L'assertion Π(r) = 0 revient essentiellement à dire, en terminologie plus classique, que r est modulaire (ou automorphe). Le cas où F = Q et G = GL 2 est connu et découle de la conjecture de modulairité de Serre prouvée par Khare-Wintenberger dans [START_REF] Khare | On Serre's conjecture for 2-dimensional mod p representations of Gal(Q/Q)[END_REF]. Malheureusement (i) reste toujours non résolu en général et dans cette thèse il sera parfois nécessaire de le mettre dans l'hypothèse. On suppose que (i) est vraie et on considère la question (ii). La réponse complète à (ii) est connue dans le cas où F = Q, G = GL 2 , elle est dû à Emerton (qui repose sur l'existence de pLL pour GL 2 (Q p )) mais est toujours ouverte quand K = Q p ou quand n ≥ 3. Un des obstacles principaux est l'absence d'une classification complète des représentations lisses irréductibles de GL n (K) à coefficients dans k E (sauf le cas GL 2 (Q p ) qui est connu d'après [START_REF] Barthel | Irreducible modular representations of GL 2 of a local field[END_REF] et [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) I[END_REF]). Plus précisément, les résultats de [START_REF] Breuil | Towards a modulo p Langlands correspondence for GL 2 Memoirs of[END_REF], [START_REF] Hu | Sur quelques représentations supersingulières de GL 2 (Q p f )[END_REF] et [START_REF] Schraen | Sur la présentation des représentations supersingulières de GL 2 (F )[END_REF] montrent que la classification des représentations qui n'apparaissent pas comme sous-quotients d'inductions paraboliques (elles sont appelées supercuspidales) est très difficile même pour GL 2 (Q p 2 ). En revanche, puisque l'on sait comment classifier toutes les représentations paraboliquement induites par les caractères d'un sous-groupe de Borel (voir [START_REF] Herzig | The classification of irreducible admissible mod p representations of a p-adic GL n[END_REF] pour les cas GL n (K)), on peut déjà construire en utilisant [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF] une représentation lisse Π ord (ρ) sur k E d'une longueur finie (qui ne dépend que de ρ) qui se plonge toujours dans Π(r) si K = Q p et si ρ est ordinaire (i.e. ρ a son image dans un sous-groupe de Borel de GL n (k E )). Mais il est clair, compte tenue de la construction dans [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF], que Π ord (ρ) n'est pas suffisant pour déterminer ρ en général. Notons que Hu montre dans [START_REF] Hu | An application of a theorem of Emerton to mod p representations of GL 2[END_REF] qu'une représentations supercuspidale apparaît nécessairement dans Π(r) même si

K = Q p 2 et n = 2.
Par conséquent, on a besoin d'une méthode qui nous permet de trouver suffisament d'informations dans Π(r) pour déterminer ρ. Une approche naturelle est de restreindre l'action de GL n (K) à GL n (O K ). Il y a au moins deux raisons pour faire ceci : les représentations irréductibles de GL n (O K ) à coefficients dans k E (donc de GL n (k)) sont faciles à classifier grace à la section 2 de la partie II de [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF]; au moins la classification des représentations irréductibles de GL n (k) à coefficients dans E est bien connue d'après la théorie de Deligne-Lusztig de [START_REF] Deligne | Representations of reductive groups over finite fields[END_REF]. En plus on a Théorème 1.2.1 ( voir [START_REF] Jantzen | Zur Reduktion modulo p der Charaktere von Deligne und Lusztig[END_REF] et la Proposition 4.1.3 de [START_REF] Le | Weight elimination in Serre type conjectures[END_REF] ). Pour toute représentation de Deligne-Lusztig τ suffisament générique et pour toute O E -réseau τ • τ , L'ensemble des facteurs de Jordan-Hölder

JH GLn(k) (τ • ⊗ O E k E )
est indépendant du choix de τ • et admet une description purement combinatoire via le groupe de Weyl affine de GL n / k .

Ici par suffisamment générique on veut dire que, quand on paramétrise une représentation de Deligne-Lusztig par un poids et un élément du groupe de Weyl comme dans [START_REF] Jantzen | Zur Reduktion modulo p der Charaktere von Deligne und Lusztig[END_REF] (voir aussi la Section 4 de [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF]), ce poids doit être suffisamment éloigné du bord de l'alcôve qui le contient.

La première étape pour comprendre Π(r)| GLn(O K ) consiste à caractériser le socle soc GLn(O K ) Π(r) ce qui est déjà un problème profond et qui est historiquement une reformulation de la partie poids de la conjecture de Serre lorsque F = Q et G = GL 2 . Les conjectures sur la structure de soc GLn(O K ) Π(r) sont formulée par Buzzard-Diamond-Jarvis dans [START_REF] Buzzard | On Serre's conjecture for mod Galois representations over totally real fields[END_REF] si K = Q p f et n = 2, par Herzig [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF] si K = Q p et n arbitraire, et par Gee-Herzig-Savitt [GHS] dans certains cas plus généraux. On s'attend à ce qu'il y ait une relation profonde entre la structure de soc GLn(O K ) Π(r) et les fibre spéciales de différents anneaux de déformations potentiellement semi-stables de ρ (conjecture de Breuil-Mézard-Emerton-Gee, c.f. [START_REF] Breuil | Multiplicités modulaires et représentations de GL 2 (Z p ) et de Gal(Q p /Q p ) en = p avec an appendice par G. Henniart[END_REF], [START_REF] Breuil | Multiplicités modulaires raffinées[END_REF] et [START_REF] Emerton | A geometric perspective on the Breuil-Mézard conjecture[END_REF]). Quand K = Q p f la structure de soc GLn(O K ) Π(r) a été déterminée pour n = 2 dans [GK14] et pour n = 3 dans [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], [START_REF] Morra | Serre weights for three-dimensional ordinay Galois representations[END_REF], [LMP] et [LLHLMa]. On pose K(1) := Ker (GL n (O K )

GL n (k))

et on remarque qu'il est naturel, du point de vue de la théorie des représentations, d'étudier Π(r) K(1) qui contient évidemment soc GLn(O K ) Π(r) comme une sous-représentation. La représentation Π(r) K(1) est un point de départ (déjà non-trivial) pour reconstruire Π(r) comme une représentation de GL n (K). Mais malheureusement les résultats de [START_REF] Hu | Multiplicity one for the mod p cohomology of Shimura curves: the tame case[END_REF], [START_REF] Le | Multiplicity one at full congruence level[END_REF] et [START_REF] Le | Multiplicity one for wildly ramified representations[END_REF] si K = Q p f et n = 2, ainsi que des travaux en préparation des auteurs de [LLHLMa] si K = Q p f et n = 3 suggèrent que Π(r) K(1) , comme représentation de GL n (k), est toujours insuffisant pour déterminer ρ, surtout dans le cas où ρ est ordinaire et indécomposable. On a l'exemple suivant Exemple 1.2.2. On prend K = Q p f , n = 2, et ρ une représentation galoisienne ordinaire de la forme suivante χ 2 * 0 χ 1 .

Supposons en plus que χ 1 χ -1 2 satisfait une hypothèse générique. Il s'ensuit d'un calcul standard de caractéristique d'Euler-Poincaré en cohomologie galoisienne que

dim k E Ext 1 Gal(Q p f /Q p f ) (χ 2 , χ 1 ) = f
et par conséquent ρ est déterminé par ρ ss et f -1 invariants à valeurs dans k E ∪ {∞} à isomorphisme près. On sait que ρ est Fontaine-Laffaille dans ce cas, alors on peut utiliser les modules de Fontaine-Laffaille (voir [START_REF] Fontaine | Construction de représentations p-adiques[END_REF]) pour définir un ensemble d'invariants dans k E ∪ {∞} (voir Lemme 2.1.1 de [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF]) qui détermine la classe d'isomorphisme de ρ. Il se trouve que la structure de soc GLn(O K ) Π(r) ne peut pas déterminer ces invariants (s'ils ne sont ni 0 ni ∞). Plus généralement, supposons que toutes les representations à coefficients dans k E de Gal(K/K) de dimension n peuvent être paramétrisées par un certain espace de module et qu'un tel espace admet une stratification par des sous-schémas localement fermés donnés par certaines conditions explicites, alors on s'attend à ce que soc GLn(O K ) Π(r) nous permette seulement de dire dans quelle strate ρ se trouve, au lieu de dire quel point de la strate ρ correspond précisement.

On définit le sous-groupe d'Iwahori I, resp. le pro-p sous-groupe d'Iwahori I(1), de GL n (K) comme la préimage de l'ensemble des matrices triangulaires supérieures (resp. unipotentes) de GL n (k) via la surjection GL n (O K )

GL n (k). Il est bien connu que le normalisateur de I dans GL n (K) est engendré par I et l'élément suivant:

Ξ n :=        0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • 1 K 0 0 • • • 0        ∈ GL n (K).
Puisque Ξ n et GL n (O K ) engendrent GL n (K), on s'attend naturellement à ce que Ξ n joue un rôle crucial, quelle que soit la méthode utilisée, dans la reconstruction d'une représentation lisse irréductible de GL n (K) à partir da sa restriction à GL n (O K ). Dans [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF], les auteurs ont montré que (si K = Q p f et n = 2) soc GLn(O K ) Π(r) et l'action de Ξ 2 sur Π(r) I(1) détermine ρ à isomorphisme près; c'est le premier résultat sur la détermination des valeurs des invariants de Fontaine-Laffaille (ceci est résumé dans l'Exemple 1.2.2). Le résultat de [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF] a été récemment généralisé au cas où K = Q p et n = 3 dans [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] quand ρ est ordinaire et Fontaine-Laffaille, et dans [LMP] quand ρ a une sous représentation ou un quotient irréductible de dimension deux. Ces deux approches pour n = 3 considèrent l'action de Ξ 3 sur Π(r) I(1) et requièrent des hypothèses génériques techniques supplémentaires sur ρ qui consistent essentiellement à garantir que soc GLn(O K ) Π(r) a une longueur minimale si on fixe ρ ss et si on fait varier les paramètres d'extension de ρ. La première partie de ma thèse consiste à généraliser les résultats de [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] au cas où K = Q p et n est arbitraire. Ceci est fait dans le Chapitre 3; il s'agit d'un travail en commun avec Chol Park.

Théorème 1.2.3. Supposons que Π(r) = 0. Alors la structure de répresentation lisse admissible de GL n (Q p ) à coefficients dans k E de Π(r) détermine ρ à isomorphisme près si ρ est Fontaine-Laffaille, ordinaire et suffisamment générique.

Remarquons que le cas où K = Q p f et n = 3 a récemment aussi été obtenu par Enns dans [En]. Un autre résultat important dans cette direction est le Théorème 7.8 de [START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] où Scholze a utilisé une méthode géométrique complètement différente de la nôtre pour montrer que Π(r) détermine toujours ρ pour n = 2 et K arbitraire.

Maintenant on donne un énoncé plus précis du Théorème 1.2.3 et on renvoie les lecteurs au Chapitre 3 pour plus de détails. On pose K = Q p à partir de maintenant. On fixe r tel que Π(r) = 0 et suppose que ρ est ordinaire. Alors il existe une suite de sous-représentations

0 ρ 1,1 ρ 1,2 • • • ρ 1,n-1 ρ 1,n = ρ telle que
χ 1 := ρ 1,1 and χ i := ρ 1,i /ρ 1,i-1 sont de dimension 1 pour tout 2 ≤ i ≤ n. Si on suppose que χ i-1 χ -1 i satisfait certaines hypothèses (pas très fortes) qui ressemblent à celle qui a été mentionnée dans l'Exemple 1.2.2, alors on a Hom Gal(Qp/Qp) (χ i , χ i-1 ) = Ext 2 Gal(Qp/Qp) (χ i , χ i-1 ) = 0 et dim k E Ext 1 Gal(Qp/Qp) (χ i , χ i-1 ) = 1 pour tout 2 ≤ i ≤ n. Par un dévissage on en déduit que dim k E Ext 1 Gal(Qp/Qp) (χ i , ρ 1,i-1 ) = i -1 qui dit grosso modo que ρ i est déterminé par χ i , ρ 1,i-1 et i -2 invariants supplémentaires pour tout 2 ≤ i ≤ n. Par une récurrence rapide on sait que ρ est déterminé par

ρ ss = ⊕ n i=1 χ i et (n-1)(n-2) 2
invariants. Selon une classification complète des caractères de Gal(Q p /Q p ) à valeurs dans k E via la théorie de corps de classes, on sait qu'il existe a i ∈ Z/(p -1)Z tel que

χ i ω -ai-i+1
est non ramifié pour tout 1 ≤ i ≤ n où ω est la réduction modulo p du caractère cyclotomique

ε : Gal(Q p /Q p ) Z × p .
Il se trouve que la condition que ρ est Fontaine-Laffaille est essentiellement équivalente à l'existence d'entiers a i ∈ Z dont les images dans Z/(p -1)Z sont les a i et vérifiant

a 1 + p -1 > a n > a n-1 > • • • > a 1 .
La théorie de [START_REF] Fontaine | Construction de représentations p-adiques[END_REF] associe à ρ un module de Fontaine-Laffaille FL(ρ). On peut définir un certain ensemble d'invariants FL i,j (ρ), pour toute paire d'entiers (i, j) tels que 1 ≤ i < i + 1 < j ≤ n, comme certaines fonctions rationnelles explicites en entrées de la matrice de Frobenius du module FL(ρ) et prouver que les FL i,j (ρ) ne dépendent que de la classe d'isomorphisme de FL(ρ). On observe que ρ est déterminé par ρ ss et ces invariants FL i,j (ρ), et que la démonstration du Théorème 1.2.3 se réduit au problème de détecter tous les invariants FL i,j (ρ) dans Π(r). Une idée générale consiste à étudier Π(r) via Π sm (r) pour de différents relèvements r d'un r fixé tels que les représentations galoisiennes locales ρ = r| Gal(Fw 0 /Fw 0 ) satisfassent des conditions supplémentaires. Plus précisément, pour une représentation de Deligne-Lusztig τ de GL n (F p ) donnée, il existe un type inertiel modérément ramifié LL -1 (τ ) (qui est une représentation du sous-groupe d'inertie I Qp Gal(Q p /Q p ) qui se factorise par le quotient modérément ramifié) qui lui correspond via la correspondance de Langlands locale inertielle (voir Section 3 de [CEGGPS]). La notation LL -1 (τ ) provient de la compatibilité entre la correspondance de Langlands locale classique et la correspondance de Langlands locale inertielle. Alors on considère tous les relèvements ρ de ρ avec les poids de Hodge-Tate {0, 1, • • • , n -1} tels que WD(ρ)| I Qp ∼ = LL -1 (τ ).

(1.2.4)

On remarque que tout ρ correspond à un E-point du schéma formel donné par un anneau local noethérien complet R

LL -1 (τ ),{0,••• ,n-1} ρ
. C'est un cas spécial de la construction des anneaux de déformation potentiellement semi-stables de [START_REF] Kisin | Potentially semistable deformation rings[END_REF] pour les types inertiels et les poids de Hodge-Tate plus généraux.

Pour toute paire d'entiers (i, j) tels que 1 ≤ i < i + 1 < j ≤ n, on choisit une représentation de Deligne-Lusztig τ i,j puis on prend une représentation galoisienne r i,j : Gal(F /F ) → GL n (E) telle que (i) r i,j est automorphe et non-ramifié en toute place finie de F au-dessus de v ∈ D(U v0 );

(ii) r i,j contient un O E -réseau r • i,j invariant dont la réduction modulo E est r;

(iii) ρ i,j := r i,j | Gal(Fw 0 /Fw 0 ) correspond à un morphisme R

LL -1 (τi,j ),{0,••• ,n-1} ρ → E.
En particulier on a le diagramme commutatif suivant:

Π sm (r i,j ) Π(r i,j ) S(U v0 , O E )[p r • i,j ] S(U v0 , O E )[p r • i,j ] Π(r i,j ) τ i,j τ • i,j τ i,j ? O O _ ? o o ? O O _ ? o o / / ? O O ? O O _ ? o o ? O O / / / / (1.2.5) où τ • i,j := τ i,j ∩ S(U v0 , O E )[p r • i,j ] Π sm (r i,j ) et τ i,j := τ • i,j ⊗ O E k E .
En prenant les sous-espaces invariants par I(1), le diagramme (1.2.5) induit un autre diagramme:

Π sm (r i,j ) I(1) S(U v0 , O E )[p r • i,j ] I(1) Π(r i,j ) I(1) τ I(1) i,j

(τ • i,j ) I(1) τ i,j I (1)
. i,j = 0 implique que τ i,j est une représentation de la série principale de GL n (F p ). Cela revient à dire que τ i,j provient d'une induction parabolique d'un caractère de T (F p ). Chaque terme de (1.2.6) admet une action de l'algèbre de Hecke-Iwahori qui contient Ξ n et n -1 opérateurs U m n ∈ Z p [GL n (Q p )] pour 1 ≤ m ≤ n -1. On note P m le parabolique standard de GL n qui contient le sous-groupe de Borel triangulaire supérieur et qui a des blocs de Levi de la forme GL m × GL n-m . On note N m le radical unipotent de P m . Le lemme suivant résume les deux propriétés principales de U m n .

_
Lemme 1.2.7. On a

(Ξ n ) m • U m n ∈ Z p [GL n (Z p )].
Etant donnée une représentation irréductible lisse Π m (resp. Π n-m ) de GL m (Q p ) (resp. de GL n-m (Q p )) dont le caractère central est ω Πm (resp. ω Πn-m ), on a

U m n = ω Πm (p) -1
restreint à l'image de

Π m ⊗ E Π n-m → Ind GLn(Qp) Pm(Qp) (Π m ⊗ E Π n-m ) Nm(Zp)
.

On note µ i,j un caractère T (F p ) → Z × p (à déterminer plus tard) et µ i,j : T (F p ) → F × p sa réduction modulo p. Le groupe T (F p ) agit naturellement sur l'espace Π sm (r i,j ) I(1) . On note Π sm (r i,j ) I(1),µi,j le sous espace propre associé au caractère µ i,j . Étant donnée une valeur propre α ∈ E × de U m n sur Π sm (r i,j ) I(1),µi,j , on cherche un diagramme:

interpréter α via une identité dans Zp[GLn(Qp)] sur (τ • i,j ) I(1),µ i,j τ I(1),µ i,j i,j

Π sm (ri,j ) I(1),µ i,j

interpréter FLi,j (ρ) via une identité dans Fp[GLn(Qp)] sur τi,j I(1),µ i,j Π(ri,j ) I(1),µ i,j α montrer que α est un produit de valeurs propres du Frobenius de WD(ρi,j )

FL i,j (ρ) étape 1 étape 2 étape 3 étape 4 k s k s + 3
(1.2.8) qui a déjà apparu dans [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF]. On insiste sur le fait que k est un entier qui vérifie 1 ≤ k ≤ n -1 et qu'il est déterminé par la paire (i, j). La partie la plus standard du diagramme 1.2.8 est l'étape 3 qui est essentiellement une égalité entre valeurs propres de U m n et produits de certaines valeurs propres du Frobenius de WD(ρ i,j ) et qui découle directement de la correspondance de Langlands locale classique et de la compatibilité local-global classique (voir Théorème 1.1.1). L'étape 4 qui réalise l'invariant FL i,j (ρ) comme la reduction modulo E d'un produit de valeurs propres du Frobenius (multiplié par une puissance convenable de p), est prouvée par des calculs techniques de la théorie de Hodge p-adique entière, notamment via des modules de Breuil et de Kisin. La plupart de l'étape 4 est faite dans Section 3.3. Il est nécessaire d'insister sur une différence importante entre l'étape 3 et l'étape 4. L'étape 3 situe complètement en caractéristique 0 à coefficients dans E, ne dépend que de WD(ρ) et n'exige pas de conditions supplémentaires sur ρ. En revanche l'étape step 4 repose sur une condition technique supplémentaire sur ρ qui sera rappelée dans la Définition 1.2.9.

On peut associer à une paire (ρ, τ i,j ) un élément w(ρ, τ i,j ) du groupe de Weyl affine W de GL n en utilisant la théorie des shapes qui est essentiellement développée dans [LLHLMa]. Pour toute représentation galoisienne semi-simple

ρ 0 : Gal(Q p /Q p ) → GL n (k E ) on définit l'ensemble suivant Ω(ρ 0 , τ i,j ) := { w(ρ, τ i,j ) | ρ ss ∼ = ρ 0 }.
On considère la longueur ( w(ρ, τ i,j ))

par rapport au système de Coxeter standard de W .

Définition 1.2.9. On dit que ρ est τ i,j -générique si la longueur de w(ρ, τ i,j ) est maximale parmi celles des éléments de Ω(ρ ss , τ i,j ).

On peut construire (via les modules de Fontaine-Laffaille) un k E -schéma M ρ 0 dont les k E -points paramétrisent toute la représentation de Fontaine-Laffaille ρ avec une semi-simplification fixée ρ ss ∼ = ρ 0 . On a alors une stratification naturelle S de M ρ 0 indexée par l'ensemble Ω(ρ 0 , τ i,j ) (ceci découle des travaux en cours des auteurs de [LLHLMa] sur l'espace de module des modules de Kisin avec données de descente modérément ramifiées). Il s'avère que M ρ 0 est irréductible et il existe un unique élément de longueur maximale dans Ω(ρ 0 , τ i,j ) qui correspond à l'unique strate ouverte (non vide) dans S , ce qui implique que τ i,j -générique est en effet une hypothèse générique.

Il découle du Lemme 1.2.7 que U m n = α i,j (1.2.10) sur Π sm (r) I(1),µi,j pour une certaine constante α i,j ∈ E × . On pose h i,j := val p (α i,j ), α i,j := p -hi,j α i,j ∈ O × E et on note α i,j la réduction modulo E de α i,j . Le résultat principal issu des étapes 3 et 4 en (1.2.8) est:

Proposition 1.2.11. On a l'égalité

α i,j = FL i,j (ρ) -1 ∈ k × E (1.2.12)
si ρ est τ i,j -générique.

Il reste à clarifier les étapes 1 et 2 en (1.2.8). Il nous faut deux opérateurs

S i,j , S i,j, ∈ Z p [GL n (F p )] et leurs réductions modulo p S i,j , S i,j, ∈ F p [GL n (F p )]
qui ne dépendent que du choix de τ i,j et de µ i,j . Alors l'égalité dans Z p [GL n (Q p )] dont on a besoin est de la forme

S i,j, • (Ξ n ) m • U m n = c i,j S i,j
(1.2.13) (sur Π sm (r) I(1),µi,j ) où c i,j ∈ E × est une constante qui ne dépend que du choix de τ i,j , µ i,j et k. On insiste que l'égalité (1.2.13) peut être calculé dans

Z p [GL n (Z p )] parce que (Ξ n ) m • U m n ∈ Z p [GL n (Z p )]
d'après le Lemme 1.2.7. On suppose que

p -hi,j c i,j ∈ O × E
et on note c i,j les réductions modulo E de p -hi,j c i,j . Alors en combinant (1.2.13) et (1.2.10), on obtient:

S i,j, • (Ξ n ) m = α -1 i,j c i,j S i,j (1.2.14)
sur Π sm (r) I(1),µi,j dont les réductions modulo E sont

S i,j, • (Ξ n ) m = c i,j FL i,j (ρ)S i,j (1.2.15)
sur Π(r) I(1),µi,j par la Proposition 1.2.11, quitte à faire l'hypothèse supplémentaire suivante:

Hypothèse 1.2.16. On a dim k E Π(r) I(1),µi,j = 1 et τ i,j I(1),µi,j ∼ -→ Π(r) I(1),µi,j .

On observe que l'Hypothèse 1.2.16 exige un choix optimal du niveau U v0 . Sinon on aurait besoin de la remplacer par une autre hypothèse qui assure qu'un certain module sur l'algèbre de Hecke T est libre sur son support. Pour supprimer l'Hypothèse 1.2.16, on a besoin d'une version forte de la condition τ i,j -générique. En fait, il découle d'un calcul des modules de Kisin avec données de descente modérément ramifiées (qui généralise les calculs de [LLHLMa] et sera fait dans [LLMPQ]) qu'il existe un sous-schéma ouvert non vide M sm ρ 0 de M ρ 0 contenu dans une strate ouverte τ i,j -générique tel que

ρ est un k E -point de M sm ρ 0 si et seulement si R LL -1 (τi,j ),{0,••• ,n-1} ρ
est formellement lisse. L'existence même de M sm ρ 0 donne une restriction très forte sur le choix des τ i,j .

Définition 1.2.17. On dit que ρ est fortement τ i,j -générique s'il est donné par un k E -point de M sm ρ 0 .

La proposition suivante (qui est très liée à la conjecture de Breuil-Mézard-Emerton-Gee) sera prouvée dans [LLMPQ] et découle essentiellement d'un foncteur construit par la méthode du patching de Taylor-Wilies-Kisin (voir la Section 7.3 de [LLHLMa] pour le cas de GL 3 (Q p f )).

Proposition 1.2.18. L'Hypothèse 1.2.16 est vraie pour un choix convenable de U v0 si ρ est fortement τ i,j -générique.

Il faut comprendre pourquoi les opérateurs S i,j , S i,j, ∈ Z p [GL n (F p )] (satisfaisant (1.2.13)) existent et s'il est possible de les construire explicitement et de calculer la constante c i,j . L'existence de certains S i,j et S i,j, qui satisfont (1.2.13) est relativement formelle et découle essentiellement de l'Hypothèse 1.2.16 mais les opérateurs peuvent être très méchants et par conséquent on ne peut rien dire sur la constante c i,j . En revanche la construction explicite d'une seule paire de tels opérateurs est très technique et la partie la plus difficile est de montrer que S i,j et S i,j, ne s'annulent pas sur Π(r) I(1),µi,j . Dans le Chapitre 3, on donne en effet (avec des notations différentes) une construction explicite de ces opérateurs, on prouve la non-annulation mentionnée ci-dessus dans la Section 3.4.8 et on calcule la constante c i,j dans la Section 3.4.5.

On conclut que l'identité (1.2.15) montre que l'invariant FL i,j (ρ) apparaît dans Π(r) si ρ est fortement τ i,j -générique, et que Π(r) détermine ρ à isomorphisme près si ρ est fortement τ i,j -générique pour toute paire (i, j) telle que 1

≤ i < i + 1 < j ≤ n.
Il y a plusieurs généralisations naturelles du Théorème 1.2.3 qui ne sont pas incluses dans cette thèse. On en donne un résumé rapide. On suppose dans la suite que ρ ss est suffisamment générique. Les généralisations suivantes sont pour la plupart des travaux en cours avec Viet Bao Le Hung, Daniel Le, Stefano Morra et Chol Park.

La première étape est de supprimer la condition ordinaire. La nouvelle difficulté se divise en deux parties : (i) on a besoin d'une famille d'invariants de Fontaine-Laffaille et d'un convenable τ pour tout invariant afin que la stratégie en (1.2.8) s'applique; (ii) il faut construire explicitement certaines paires d'opérateurs S, S ∈ Z p [GL n (F p )] pour tout invariant afin qu'un analogue de (1.2.13) soit vrai pour une constante explicite c. On remarque que il y a beaucoup plus de cas de ρ. La seconde étape est de éliminer certaines conditions τ -générique qui sont analogues à la Définition 1.2.9. La difficulté de cette étape du fait que R

LL -1 (τ ),{0,••• ,n-1} ρ
ait peu de chance d'être formellement lisse sans une hypothèse générique forte sur ρ. Ainsi la Proposition 1.2.11, l'Hypothèse 1.2.16 et (1.2.15) s'écroulent complètement en général. L'observation cruciale est que l'on peut déterminer ρ en utilisant un ensemble d'invariants et un ensemble de poids de Serre modulaires. La dernière étape consiste à éliminer la condition Fontaine-Laffaille. Le problème est que ρ admet d'habitude des relèvements pour beaucoup moins de choix possibles de τ si on fixe les poids de Hodge-Tate. La solution naturelle est de considérer les relèvements (de ρ) dont les poids de Hodge-Tate sont plus généraux.

Remarque 1.2.19. Il serait possible de généraliser toutes les discussions ci-dessus au cas où K = Q p f en utilisant les travaux récents de Enns dans [En]. Si K est ramifié au-dessus de Q p , la situation semble beaucoup plus compliquée parce que Π(r) K(1) est en général trop petit et que l'on aurait besoin de types inertiels sauvagement ramifiés pour trouver suffisamment d'invariants de ρ dans Π(r). Le seul résultat connu sans restriction sur la ramification de K est dû à Scholze [START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] pour n = 2 et K arbitraire où une méthode complètement différente et plus géométrique est employée. L'auteur ne sait pas comment généraliser la méthode de [START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] au cas n arbitraire pour déterminer ρ.

Dilogarithme et L -invariants supérieurs pour GL

3 (Q p )
Maintenant on passe à la représentation localement analytique Π an (r) qui est l'objet central de la seconde partie de ma thèse. On peut utiliser la théorie des modules de Verma (autrement dit la théorie de catégorie de Bernstein-Gelfand-Gelfand O expliquée par exemple dans [START_REF] Humphreys | Representations of Semisimple Lie Algebras in the BGG Category O[END_REF]) pour étudier Π an (r). Très peu est connu sur la classification complète d'objets irréductibles dans la catégorie Rep la GLn(K),E des représentations localement analytiques admissibles de GL n (K). On considère la sous-catégorie pleine Rep PS GLn(K),E (qui est beaucoup mieux comprise) de Rep la GLn(K),E engendré par les sous-quotients des séries principales localement analytiques Ind GLn(K) B(K) χ an pour χ un caractère localement analytique arbitraire de T (K). Quand on dit qu'une sous-catégorie pleine est engendrée par une famille d'objets, on veut dire qu'elle est la sous-catégorie pleine minimale (tout objet est de longueur finie) qui contient cette famille d'objets et qui est stable par extensions et sous-quotients. On suppose que r est non ramifié en toute place finie de F au-dessus de D(U v0 ) et que Π alg (r) = 0.

(1.

JH PS GLn(K),E (r) ⊆ JH PS GLn(K),E
comme le sous-ensemble des objets irréductibles Π irr tels qu'il existe un plongement Π irr → Π an (r).

On définit la multiplicité:

d Π irr := dim E Hom GLn(K) Π irr , Π an (r)
qui est finie pour tout Π irr ∈ JH PS GLn(K),E (r) puisque Π an (r) est admissible. Si ρ est crystalline et satisfait une certaine hypothèse de généricité (faible), alors l'ensemble JH PS GLn(K),E (r) est explicitement déterminé par certaines données combinatoires données par la filtration de Hodge de ρ par [START_REF] Breuil | A local model for the trianguline variety and applications[END_REF]. Sous les mêmes conditions, [START_REF] Breuil | Towards the finite slope part for GL n , prépublication[END_REF] construit une certaine sous-représentation de Π an (r) qui est maximale parmi toutes les représentations qui satisfont une certaine condition de multiplicité une et qui appartiennent à Rep PS GLn(K),E . Si ρ est crystalline, les résultats de la Section 3.3 de [Bre17] et de [START_REF] Breuil | Towards the finite slope part for GL n , prépublication[END_REF] suggèrent que la question de comment construire une sous-représentation suffisamment large de Π an (r) pour déterminer complètement ρ est encore mystérieux (surtout quand la filtration de Hodge de ρ se trouve dans une position suffisamment générique). La situation est assez différente (et meilleure) si ρ est semi-stable et non-crystalline, au moins quand n = 3. On suppose à partir de maintenant que

F + = Q (ainsi K = Q p et v 0 = p) et que le (ϕ, N )-module D associé à WD(ρ) a la forme suivante: D = Ee 0 ⊕ Ee 1 ⊕ Ee 2 (1.3.2) où N (e 0 ) = 0, N (e 1 ) = e 0 , N (e 2 ) = e 1 et ϕ(e i ) = ur(α)p i-2 e i pour 0 ≤ i ≤ 2 et un certain α ∈ E × . Il découle de la correspondance de Langlands locale classique que LL • WD(ρ) = St ∞ 3 ⊗ E (ur(α) • det 3 ) où St ∞
3 est la représentation de Steinberg lisse de GL 3 (Q p ) définie comme l'unique quotient irréductibles de la série principale lisse:

Ind GL3(Qp) B(Qp) 1 ∞ , det 3 est le caractère déterminant GL 3 (Q p ) Q × p et ur(α) est le caratère non ramifée qui envoie p à α. On note k = (k 1 , k 2 , k 3 ) ∈ Z 3 avec k 1 > k 2 > k 3
l'ensemble des poids de Hodge-Tate de ρ. On définit

Π alg (k, D) := L(λ) ⊗ E St ∞ 3 ⊗ E (ur(α) • det 3 )
où L(λ) est la représentation algébrique de dimension finie de GL 3 (Q p ) dont le poids le plus haut est

λ = (λ 1 , λ 2 , λ 3 ) = (k 1 -2, k 2 -1, k 3 ) (1.3.3)
par rapport au sous-groupe de Borel triangulaire supérieur B(Q p ). Le Théorème 1.1.1 implique l'égalité suivante: Π alg (r) = Π alg (k, D) ⊕d pour un entier d = d Π alg (k,D) qui dépend de U p et de r. On dit que ρ est non-critique (voir le (ii) de la Remarque 6.1.4 de [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]) si la filtration de Hodge de ρ satisfait une hypothèse générique explicite qui implique l'égalité suivante:

JH PS GLn(K),E (r) = {Π alg (k, D)}. (1.3.4)
Si ρ est non-critique, alors il existe un triplet d'invariants (L i,j ) 0≤i<j≤2 ∈ E 3 tel que la filtration de Hodge (faiblement admissible) Fil • sur D a la forme suivante:

Fil i D =        D si i ≤ k 3 E(e 2 + L 1,2 e 1 + L 0,2 e 0 ) ⊕ E(e 1 + L 0,1 e 0 ) si k 3 + 1 ≤ i ≤ k 2 E(e 2 + L 1,2 e 1 + L 0,2 e 0 ) si k 2 + 1 ≤ i ≤ k 1 0 si i ≥ k 1 + 1 .
(1.3.5) Dans l'étude de pLL l'un des objectifs principaux est de trouver dans le cas (1.3.5) tous les invariants L i,j dans Π an (r) pour 0 ≤ i < j ≤ 2. On appelle L -invariants ces L i,j et L -invariants simples les L 0,1 et les L 1,2 . On appelle L 0,2 le L -invariant supérieur de ρ. Ce qui fait ce cas (1.3.5) si spécial est le fait que l'on espère trouver tous les invariants L i,j dans une sous-représentation de Π an (r) qui appartient à Rep PS GLn(K),E . Le cas de GL 2 (Q p ) a été étudié en premier par Breuil dans [START_REF] Breuil | Invariant L et série spéciale p-adique[END_REF] et dans [START_REF] Breuil | Série spéciale p-adique et cohomologie étale complétée[END_REF], et c'était l'un des points de départ de pLL. Afin de retrouver explicitement l'unique L -invariant de Π an (r) quand ρ est semi-stable non-crystalline de dimension deux, on doit considérer la fonction logarithme p-adique pour construire une base d'un certain groupe de Ext 1 . Un résultat récent de Ding dans [START_REF] Ding | Simple L-invariants for GL n[END_REF] généralise [START_REF] Breuil | Invariant L et série spéciale p-adique[END_REF] et [START_REF] Breuil | Série spéciale p-adique et cohomologie étale complétée[END_REF] à tous les L -invariants simples (qui sont les généralisations de L 0,1 et L 1,2 ) de GL n (K) pour K arbitraire. Il reste donc à traiter L 0,2 . Dans [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF], Schraen a étudié quelques représentations localement analytiques de GL 3 (Q p ) pour essayer de réaliser le module filtré Fil • D dans le complexe de de Rham de l'espace de Drinfeld de dimension deux. L'un des ingrédients principaux de [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] est un calcul des groupes de cohomologie localement analytiques. Schraen a construit un objet Σ(λ, L ) (voir la Définition 5.19 de [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]) dans la catégorie dérivée D Mod D(GL3(Qp),E) de la catégorie abélienne Mod D(GL3(Qp),E) des modules abstraits sur l'algèbre de distribution localement analytique D(GL 3 (Q p ), E). La motivation de cette construction est de chercher une famille raisonnable de représentations localement analytiques de GL 3 (Q p ) qui dépend de trois invariants, puisque ρ dépend de trois invariants. Schraen construit d'abord une famille de représentations localement analytiques Σ(λ, L , L ) (voir (1.11) de [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]) qui dépendent de deux invariants. Puis il observe qu'un troisième invariant se trouve dans le groupe

Ext 2 GL3(Qp) (L(λ), Σ(λ, L , L )) (1.3.6) mais pas dans Ext 1 GL3(Qp) (L(λ), Σ(λ, L , L ))
. Il semble donc naturel dans un premier temps de remplacer une famille de représentations localement analytiques par une famille de complexes Σ(λ, L ) . Grosso modo, on espère rajouter L(λ) pour obtenir le troisième invariant mais on n'arrive à le faire qu'au sens d'un complexe abstrait dans [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. Une observation cruciale de Breuil est que l'on peut peut-être étendre chaque Σ(λ, L , L ) en une représentation localement analytique plus grande qui contient plus d'invariants, en ajoutant d'autres représentations que L(λ). Breuil a construit dans [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]

deux représentations localement analytiques Π 1 (k, D) et Π 2 (k, D) dans Rep PS GL3(Qp),E qui ne dépendent que de k et D telles que dim E Ext 1 GL3(Qp) Π i (k, D), Π alg (k, D) = 3 (1.3.7)
pour i = 1, 2. On a besoin des conditions suivantes:

Condition 1.3.8. On suppose que U p et r satisfont:

(i) Π alg (r) = 0;
(ii) WD(ρ) est donné par le cas n = 3 de (1.3.2);

(iii) ρ est non-critique;

(iv) la représentation automorphe correspondante à r n'apparaît qu'une fois dans l'espace des formes automorphes sur G(A ∞ F + ).

Alors il est montré dans le Théorème 6.2.1 dans [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] que pour toute paire fixée (U p , r) qui satisfait la Condition 1.3.8, il existe une unique représentation localement analytique Π de GL 3 (Q p ) de la forme

Π alg (k, D) Π 1 (k, D) Π 2 (k, D) (1.3.9) telle que Hom GL3(Qp) Π ⊗ E ε 2 • det 3 , Π an (r) ∼ -→ Hom GL3(Qp) Π alg (k, D) ⊗ E ε 2 • det 3 , Π an (r) . (1.3.10)
Ici on utilise la notation suivante: V W pour une extension non-scindée de W par V . Une observation cruciale est que Π dépend de quatre invariants selon (1.3.7) mais que ρ ne dépend que de trois invariants selon l'égalité (1.3.5). L'isomorphisme (1.3.10) implique l'existence d'une unique représentation Π de la forme (1.3.9) qui apparaît comme une sous-représentation de Π an (r). Puisque ρ ne dépend que trois L -invariants quand n = 3, on ne s'attend pas à ce que toutes les représentations localement analytiques de la forme (1.3.9) se plongent dans Π an (r) pour certains U p et r. Une question naturelle que l'on se pose est la suivante: Question 1.3.11. Y-a-t-il un critère purement de théorie de représentation qui détermine si une représentation de la forme (1.3.9) se plonge dans un certain Π an (r)? Une observation cruciale est que L(λ) est exactement l'critère que l'on voulait dans la Question 1.3.11 pour réduire le nombre d'invariants de Π. Plus précisément, le fait qu'une telle représentation localement analytique Π soit incluse dans une représentation plus grande avec L(λ) dans le cosocle (avec multiplicité deux) donne un critère pour que Π se plonge dans la cohomologie complétée. Une remarque importante de [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] est que la fonction dilogarithme p-adique apparaît dans l'espace de dimension deux (1.3.6) (voir la Corollaire 5.17 de [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]). Par conséquent, Breuil a suggéré qu'il devait y avoir une relation explicite entre [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF], [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] et [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF], plus précisément la fonction dilogarithme p-adique devrait être reliée à l'unique L -invariant supérieur L 0,2 de ρ, et cette relation devrait être analogue à celle entre la fonction logarithme p-adique et les L -invariants simples. On arrive donc naturellement à la question suivante: Question 1.3.12. Quelle est la relation précise entre la fonction dilogarithme p-adique et l'unique L -invariant supérieur L 0,2 de ρ?

Le résultat principal de la seconde partie de ma thèse est une tentative de répondre à ces deux questions à la fois, en utilisant la construction d'une famille de représentations localement analytiques Σ min,+ (λ, L 1 , L 2 , L 3 ) qui dépend de trois invariants L 1 , L 2 , L 3 ∈ E (voir le Théorème 4.1.1 pour une description intuitive de la représentation). L'exposant 'min, +' signifie que Σ min,+ (λ, L 1 , L 2 , L 3 ) est légèrement plus grande que la représentation localement analytique minimale de GL 3 (Q p ) qui "contient" la fonction dilogarithme p-adique. On remarque que λ et k se déterminent l'un l'autre par (1.3.3). Les propriétés principales de Σ min,+ (λ, L 1 , L 2 , L 3 ) sont résumées dans les trois énoncés ci-dessous.

Théorème 1.3.13. Etant donnés U p et r qui satisfont la Condition 1.3.8, il existe un unique choix de paramètres L 1 , L 2 , L 3 ∈ E tel que

Hom GL3(Qp) Σ min,+ (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det 3 , Π an (r) ∼ -→ Hom GL3(Qp) Π alg (k, D) ⊗ E ε 2 • det 3 , Π an (r) . (1.3.14) Remarque 1.3.15. On sait que L 1 = L 1,2 , L 2 = L 0,1 (
Π(D) → Σ min,+ (λ, L 1 , L 2 , L 3 ) (1.3.17) pour certains L 1 , L 2 , L 3 ∈ E uniquement déterminés par ρ où Π(D)
est la représentation localement analytique construite à partir de ρ dans [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF]. Mais on ne sait pas comment le prouver en utilisant une méthode purement locale. Si ρ est ordinaire (autrement dit si elle a des poids de Hodge-Tate consécutifs), alors le Théorème 1.1 de [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] prouve que Π(D) se plonge en fait dans Π an (r), ce qui avec notre Théorème 1.3.13 impliquent que (1.3.17) existe et que L 3 ne dépend que de ρ.

On donne ci-dessous une réponse positive à la Question 1.3.11 (au moins dans une direction). Elle découle facilement du Théorème 6.2.1 de [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] et de notre Théorème 1.3.13.

Corollaire 1.3.18. On fixe une représentation Π de la forme (1.3.9). Si l'on a un plongement

Π ⊗ E ε 2 • det 3 → Π an (r)
pour un choix convenable de U p et de r, alors on a un plongement

Π → Σ min,+ (λ, L 1 , L 2 , L 3 ) ⊗ E ur(α) • det 3 pour certains L 1 , L 2 , L 3 ∈ E uniquement déterminés par Π.
Chapter 2

General Introduction

p-adic local-global compatibility conjecture

We fix a prime number p, a finite extension [K : Q p ] < +∞ and another sufficiently large finite extension [E : Q p ] < +∞. We use the notation O K (resp.

K , resp. k) for the ring of integer (resp. a uniformizer, resp. the residual field) of K. Similarly, We use the notation O E (resp. E , resp. k E ) for the ring of integer (resp. a uniformizer, resp. the residual field) of E. The socalled p-adic local Langlands correspondence initiated by Breuil in [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) I[END_REF], [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) II[END_REF] and [START_REF] Breuil | Invariant L et série spéciale p-adique[END_REF] is conjecturally a bijection between a certain set of p-adic Banach linear representations of GL n (K) and the set of p-adic continuous Galois representations ρ : Gal(K/K) → GL n (E). We use the notation pLL for the map from ρ to the conjectural p-adic Banach representation pLL(ρ) corresponding to it. The full p-adic local Langlands correspondence is known for GL 2 (Q p ) essentially due to Colmez in [START_REF] Colmez | Représentations de GL 2 (Q p ) et ( , Γ)-modules[END_REF], and its compatibility with certain p-adic interpolation of étale cohomology (the so-called completed cohomology) of tower of modular curves is known by Emerton in [START_REF] Emerton | A local-global compatibility conjecture in the p-adic Langlands programme for GL 2 /Q Pure and Applied Math[END_REF]. The map pLL is largely mysterious for K = Q p or n ≥ 3 and only a few partial results are known. It is widely expected (c.f. [START_REF] Breuil | The emerging p-adic Langlands programme[END_REF]) that pLL should be compatible with reduction modulo E , 'p-adic families' and so on, hence we use the notation pLL for the map (which is highly conjectural if K = Q p or n ≥ 3) that associate a admissible smooth k E -representation of GL n (K) with a Galois representation ρ : Gal(K/K) → GL n (k E ).

We start to give more precise formulation of the conjectural local-global compatibility for pLL, which is a natural generalization of the ideas in [START_REF] Emerton | A local-global compatibility conjecture in the p-adic Langlands programme for GL 2 /Q Pure and Applied Math[END_REF] (except that we no longer know pLL if K = Q p or n ≥ 3). From now on we fix a totally real field F + together with a totally imaginary quadratic extension F of F + . We fix a finite place v 0 of F + that divides p, splits in F and satisfies K ∼ = F + v0 ∼ = F w0 where w 0 is a fixed finite place of F above v 0 . We fix a unitary group G over F + such that

(i) G ⊗ F + F ∼ = GL n / F ; (ii) G(F + ⊗ Q R) is compact. We also fix an open compact subgroup U v0 ⊂ G(A ∞,v0
F + ) and a finitely generated O E -module A endowed with its p-adic topology. Then we consider the space of p-adic (resp. locally constant) A-valued automorphic forms on G(A ∞ F + ) with a fixed level U v0 prime to v 0 :

S(U v0 , A) := {f : G(F + )\G(A ∞ F + )/U v0 → A, continuous } (resp. S(U v0 , A) := {f : G(F + )\G(A ∞ F + )/U v0 → A, locally constant }).
Note in particular that if A is a p-torsion module and has discrete topology, then the two spaces defined above coincide. Each space above carries an action of a universal Hecke algebra T over O E (freely generated by Hecke operators indexed by finite places of F lying above a set D(U v0 ) of finite places of F + which splits in F , are prime to p and where

U v := U v0 ∩G(F + v ) is a maximal open compact subgroup of G(F + v ))
, as well as an action of GL n (K) which is continuous (resp. smooth) on S(U v0 , A) (resp. on S(U v0 , A)) and comes from right translation on G(F + )\G(A ∞ F + )/U v0 ). The action of GL n (K) on both spaces commute with the action of T. Given a p-adic continuous Galois representation r A : Gal(F /F ) → GL n (A) which is absolutely irreducible, conjugate self-dual and unramified at each place of F lying above a certain v ∈ D(U v0 ), we can associate an ideal p r A ⊂ T ⊗ O E A with r A , then the torsion subspace S(U v0 , A)[p r A ] (resp. S(U v0 , A)[p r A ]) cut out by this ideal naturally carries a continuous (resp. smooth) action of GL n (K). We use the notation LL for the map given by classical local Langlands correspondence that sends an n-dimensional Frobenius semi-simple Weil-Deligne representation to its corresponding irreducible smooth representation of GL n (K) (c.f. [HT01], [START_REF] Henniart | Une preuve simple des conjectures de Langlands pour GL n sur un corps p-adique[END_REF] and [START_REF] Scholze | The Local Langlands Correspondence for GL n over p-adic fields Invent[END_REF]). We set

r := r O E ⊗ O E E, r := r k E = r O E ⊗ O E k E , ρ := r| Gal(K/K) , ρ := r| Gal(K/K)
and use the shorten notation

Π(r) := S(U v0 , O E )[p r O E ] ⊗ O E E Π(r) := S(U v0 , k E )[p r k E ] Π sm (r) := S(U v0 , O E )[p r O E ] ⊗ O E E .
Note that we omit the level U v0 from the notation for simplicity. We define Π alg (r) as the subspace of Π(r) consisting of locally algebraic vectors and note that Π alg (r) is naturally a locally algebraic representation of GL n (K). We use the notation WD(ρ) (resp. Alg(ρ)) for the map that associates a Weil-Deligne representation (resp. a finite dimensional algebraic representation of GL n (K)) to a potentially semi-stable p-adic Galois representation ρ (resp. the set of Hodge-Tate weights of ρ) via Fontaine's theory in [START_REF] Fontaine | Représentations p-adiques semi-stables[END_REF] (resp. via Section 5 of [START_REF] Breuil | Vers le socle localement analytique pour GL n I[END_REF]). At this stage, the classical local-global compatibility at p (which is proven in [START_REF] Barnet-Lamb | Local-global compatibility for = p I[END_REF] and [START_REF] Barnet-Lamb | Local-global compatibility for = p II[END_REF] with further technical assumptions) can be roughly formulated as the following Theorem 2.1.1. Assume that Π alg (r) = 0.

Then ρ is potentially semi-stable and there exists an integer d > 0 depending only on r and U v0 such that

Π alg (r) ∼ = (LL • WD(ρ) ⊗ E Alg(ρ)) ⊕d . (2.1.2)
It is not too difficult to imagine that the conjectural p-adic local-global compatibility should very roughly have the following form Hope 2.1.3. Assume that Π(r) = 0.

Then there exists an integer d > 0 depending only on r and U v0 such that

Π(r) ∼ = pLL(ρ) ⊕d (2.1.4) and Π(r) ∼ = pLL(ρ) ⊕d . (2.1.5)
In particular, Hope 2.1.3 predicts that Conjecture 2.1.6. The structure of Π(r) (resp. of Π(r)) as a p-adic admissible Banach E-representation (resp. admissible smooth k E -representation) of GL n (K) determines and depends only on the isomorphism class of ρ (resp. of ρ).

We emphasize that ρ needs not be potentially semi-stable in Hope 2.1.3 and Conjecture 2.1.6. On the other hand, as the maps pLL and pLL are highly mysterious if K = Q p or n ≥ 3, we need to start with the cases when ρ is indeed potentially semi-stable and thus Theorem 2.1.1 is available. As a result, we will essentially consider only the cases when ρ is potentially semi-stable in the rest of this introduction. It is quite difficult to study the p-adic Banach representations for GL n (K) directly, and we have essentially two kinds of possibly easier objects to start with. My thesis is thus divided into two parts and each part studies one kind of objects. The first part of my thesis studies the admissible smooth k E -representation Π(r) (which is joint work with Chol Park) while the second part of my thesis studies the locally analytic representation Π an (r) defined as the set of locally analytic vectors (vectors upon which elements of GL n (K) act via vector-valued locally analytic functions on the group GL n (K)) in Π(r).

Mod p local-global compatibility for GL

n (Q p )
We firstly start with Π(r). There are several natural questions to ask about Π(r):

(i) do we have Π(r) = 0?

(ii) what is the structure of Π(r)?

(iii) what is the explicit relation between Π(r) and ρ = r| Gal(K/K) (remember that K = F w0 )?

The statement Π(r) = 0 is essentially equivalent to the more classical terminology 'r is modular (or automorphic)'. This is known for the case F = Q and G = GL 2 as a result of Serre's conjecture which was proven by Khare-Wintenberger in [START_REF] Khare | On Serre's conjecture for 2-dimensional mod p representations of Gal(Q/Q)[END_REF]. Unfortunately, (i) is still widely open in general and therefore will be put as an assumption in my thesis whenever necessary. We assume that (i) holds and consider the question (ii). The complete answer to (ii) is known for the case F = Q, G = GL 2 due to Emerton (which relies crucially on pLL for GL 2 (Q p )) and is typically still widely open if K = Q p or n ≥ 3. One of the main obstacles is a lack of complete classification of irreducible smooth k E -representations of GL n (K) except the case GL 2 (Q p ) which is known due to [START_REF] Barthel | Irreducible modular representations of GL 2 of a local field[END_REF] and [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) I[END_REF]. More precisely, evidences from [START_REF] Breuil | Towards a modulo p Langlands correspondence for GL 2 Memoirs of[END_REF], [START_REF] Hu | Sur quelques représentations supersingulières de GL 2 (Q p f )[END_REF] and [START_REF] Schraen | Sur la présentation des représentations supersingulières de GL 2 (F )[END_REF] show that classification of representations that do not occur as subquotients of parabolic inductions (which are called supercuspidal) is very difficult even for GL 2 (Q p 2 ). On the other hand, as one knows how to classify all the representations parabolically induced from characters of a Borel subgroup (c.f. [START_REF] Herzig | The classification of irreducible admissible mod p representations of a p-adic GL n[END_REF] for general parabolic induction for GL n (K)), one can already construct by [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF] a quite big finite length smooth k E -representation Π ord (ρ) (depending only on ρ) that always embeds into Π(r) if K = Q p and ρ is ordinary (i.e. ρ has its image in a Borel subgroup of GL n (k E )). However, it is clear from the construction of [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF] that Π ord (ρ) is not sufficient to determine ρ in general. Note that Hu shows in [START_REF] Hu | An application of a theorem of Emerton to mod p representations of GL 2[END_REF] that a supercuspidal representation necessarily appears in Π(r) even in the case K = Q p 2 and n = 2 and it is widely expected that the extra information of ρ involves the extensions between Π ord (ρ) and certain supercuspidal representations.

As a result, one needs a method to 'walk around' supercuspidal representations but nevertheless to be able to find enough information in Π(r) to determine ρ. One natural approach is to restrict the action to GL n (O K ) due to at least two reasons: irreducible k E -representations of GL n (O K ) (hence of GL n (k)) are easy to classify due to section 2 in Part II of [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF]; at least classification of irreducible E-representations of GL n (k) is well-known by Deligne-Lusztig theory in [START_REF] Deligne | Representations of reductive groups over finite fields[END_REF]. Moreover, we have Theorem 2.2.1 ( c.f. [Jan81] and Proposition 4.1.3 of [START_REF] Le | Weight elimination in Serre type conjectures[END_REF] ). For each sufficiently generic Deligne-Lusztig representation τ of GL n (k) and each O E -lattice τ • τ , the set of Jordan-Hölder factors

JH GLn(k) (τ • ⊗ O E k E )
is independent of the choice of τ • and has a purely combinatorial description via the affine Weyl group of GL n / k .

Here the condition sufficiently generic means the following: when we parameterize a Deligne-Lusztig representation by a weight and an element of the Weyl group due to Jantzen (c.f. [START_REF] Jantzen | Zur Reduktion modulo p der Charaktere von Deligne und Lusztig[END_REF] and Section 4 of [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF]), this weight should be sufficiently far away from the boundary of the alcove it lies in.

The very first step to understand Π(r

)| GLn(O K ) is to characterize the socle soc GLn(O K ) Π(r)
which is already a highly non-trivial and deep problem and is historically a reformulation of the more classical weight part of Serre's conjecture if we take F = Q and G = GL 2 . Conjectures on (partial) characterizations of soc GLn(O K ) Π(r) have been made by Buzzard-Diamond-Jarvis in [START_REF] Buzzard | On Serre's conjecture for mod Galois representations over totally real fields[END_REF] for the case K = Q p f , n = 2, by Herzig in [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF] for K = Q p and general n, and by Gee-Herzig-Savitt in [GHS] for certain more general cases. It is widely expected that there is a deep relation between the structure of soc GLn(O K ) Π(r) and the special fibers of various potentially semi-stable deformations rings of ρ which finally leads us to the so-called Breuil-Mézard-Emerton-Gee conjecture formulated in [START_REF] Breuil | Multiplicités modulaires et représentations de GL 2 (Z p ) et de Gal(Q p /Q p ) en = p avec an appendice par G. Henniart[END_REF], [START_REF] Breuil | Multiplicités modulaires raffinées[END_REF] and [START_REF] Emerton | A geometric perspective on the Breuil-Mézard conjecture[END_REF].

The structure of soc GLn(O K ) Π(r) when K = Q p f was determined for n = 2 by [GK14] and for n = 3 by [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], [START_REF] Morra | Serre weights for three-dimensional ordinay Galois representations[END_REF], [LMP] and [LLHLMa]. We set

K(1) := Ker (GL n (O K ) GL n (k))
and note that it is representation theoretically natural to study Π(r) K(1) which obviously contains soc GLn(O K ) Π(r) as a subrepresentation. The representation Π(r) K(1) is a very beginning point (already highly non-trivial) to reconstruct Π(r) as a GL n (K) representation, but unfortunately various evidences from [START_REF] Hu | Multiplicity one for the mod p cohomology of Shimura curves: the tame case[END_REF], [START_REF] Le | Multiplicity one at full congruence level[END_REF] and [START_REF] Le | Multiplicity one for wildly ramified representations[END_REF] for K = Q p f , n = 2 and from work in preparation by the authors of [LLHLMa]

K = Q p f , n = 3 suggest that Π(r) K(1)
is still not sufficient to determine ρ typically in the case when ρ is ordinary and maximally non-split (i.e. has no subquotient which is direct sum of two characters). We have the following example Example 2.2.2. We take K = Q p f , n = 2 and ρ is ordinary with the following form

χ 2 * 0 χ 1 .
Assume further that χ 1 χ -1 2 satisfies a weak generic assumption, then it follows from a standard computation of Euler-Poincaré characteristic of Galois cohomology that

dim k E Ext 1 Gal(Q p f /Q p f ) (χ 2 , χ 1 ) = f
and therefore ρ is determined by ρ ss together with f -1 invariants in k E ∪ {∞} up to isomorphism. If we assume moreover that ρ is Fontaine-Laffaille (which is again a generic assumption in this setting), then we can use Fontaine-Laffaille modules (c.f. [START_REF] Fontaine | Construction de représentations p-adiques[END_REF]) to define a set of invariants in k E ∪{∞} (c.f. Lemma 2.1.1 of [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF]) that determine the isomorphism class of ρ. It turns out that the structure of soc GLn(O K ) Π(r) usually cannot determine the value of these invariants. In general, suppose that all n-dimensional k E -representation of Gal(K/K) can be parameterized by some moduli space and such space has a natural stratification by locally closed subschemes given by certain explicit conditions generalizing splitness of extensions, then soc GLn(O K ) Π(r) is expected to contain only the information of which strata ρ lies in, rather than which point rho is in the strata.

We define the so-called Iwahori subgroup I (resp. pro-p Iwahori subgroup I(1)) of GL n (K) as the pre-image of the set of upper-triangular ( resp. unipotent) matrix in GL n (k) under the surjection GL n (O K )

GL n (k). It is a standard fact in the study of p-adic reductive groups that the normalizer of I inside GL n (K) is generated by I and the following element

Ξ n :=        0 1 0 • • • 0 0 0 1 • • • 0 . . . . . . . . . . . . . . . 0 0 0 • • • 1 K 0 0 • • • 0        ∈ GL n (K).
Since Ξ n and GL n (O K ) generate GL n (K), it is natural to expect that Ξ n plays a key role in whatever possible methods to reconstruct a smooth irreducible representation of GL n (K) from its restriction to GL n (O K ).

The first result to determine the values of Fontaine-Laffaille invariants (as sketched in Example 2.2.2) is [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF] where they showed (in the case K = Q p f and n = 2) that soc GLn(O K ) Π(r) together with the action of Ξ 2 on Π(r) I(1) determine ρ up to isomorphism. The result in [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF] has recently been generalized to K = Q p and n = 3 by [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] when ρ is ordinary and Fontaine-Laffaille and by [LMP] when ρ has a two dimensional irreducible sub-(or quotient-) representation. Both results for n = 3 consider the action of Ξ 3 on Π(r) I(1) and requires further 'technical generic assumption' on ρ which essentially ensures that soc GLn(O K ) Π(r) has minimal length possible if we fix ρ ss and vary the extension parameters of ρ. The first part of my thesis is to generalize the results in [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] to K = Q p but arbitrary n in Chapter 3. This is joint work with Chol Park.

Theorem 2.2.3. Assume that Π(r) = 0. Then the structure of Π(r) as a admissible smooth k Erepresentation of GL n (Q p ) determines ρ up to isomorphism if ρ is Fontaine-Laffaille, ordinary and sufficiently generic.

Note that the case K = Q p f and n = 3 has also been worked out recently by Enns in [En]. Another important result in this direction is Theorem 7.8 of [START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] where Scholze used a completely different and more geometric method to prove that Π(r) always determines ρ for n = 2 and arbitrary K. Now we start to give a more precise statement for Theorem 2.2.3 and refer to Chapter 3 for further details. We set K = Q p from now on. We fix r such that Π(r) = 0 and assume that ρ is ordinary, then there exists a sequence of subrepresentations

0 ρ 1,1 ρ 1,2 • • • ρ 1,n-1 ρ 1,n = ρ such that χ 1 := ρ 1,1 and χ i := ρ 1,i /ρ 1,i-1
are one-dimensional for all 2 ≤ i ≤ n. Assume that χ i-1 χ -1 i satisfies some mild generic assumption similar to the one mentioned in Example 2.2.2, we have

Hom Gal(Qp/Qp) (χ i , χ i-1 ) = Ext 2 Gal(Qp/Qp) (χ i , χ i-1 ) = 0 and dim k E Ext 1 Gal(Qp/Qp) (χ i , χ i-1 ) = 1 for each 2 ≤ i ≤ n.
Therefore by a simple devissage we can deduce that

dim k E Ext 1 Gal(Qp/Qp) (χ i , ρ 1,i-1 ) = i -1
which roughly means that ρ i is determined by χ i , ρ 1,i-1 together with i -2 extra invariants for each 2 ≤ i ≤ n. Hence by an immediate induction we know that ρ is determined by

ρ ss = ⊕ n i=1 χ i together with (n-1)(n-2) 2
invariants. According to a complete classification of k E -characters of Gal(Q p /Q p ) essentially due to class field theory, we know that there exists a i ∈ Z/(p -1)Z such that

χ i ω -ai-i+1
is unramified for each 1 ≤ i ≤ n where ω is the reduction modulo p of the cyclotomic character

ε : Gal(Q p /Q p ) Z × p .
It turns out that the condition ρ is Fontaine-Laffaille is essentially the same as the existing of the integers a i ∈ Z whose images in Z/(p -1)Z is a i and satisfy

a n > a n-1 > • • • > a 1 .
The theory in [START_REF] Fontaine | Construction de représentations p-adiques[END_REF] associates a Fontaine-Laffaille module FL(ρ) with ρ, and one can define a set of invariants FL i,j (ρ) as certain explicit rational functions of the entries of the Frobenius matrix of the module FL(ρ) and prove that FL i,j (ρ) depends only on the isomorphism class of FL(ρ), for each pair of integers (i, j) satisfying 1 ≤ i < i + 1 < j ≤ n. By elementary linear algebra we know that ρ is determined by ρ ss and the invariants FL i,j (ρ), and thus Theorem 2.2.3 is reduced to the problem of capturing each invariant FL i,j (ρ) inside Π(r).

An important general philosophy is to study Π(r) via Π sm (r) for all kinds of lifts r of a fixed r such that the local Galois representation ρ = r| Gal(Fw 0 /Fw 0 ) satisfies some extra conditions. More precisely, for a given Deligne-Lusztig representation τ of GL n (F p ), there exists a tamely ramified inertia type LL -1 (τ ) (which is a representation of the inertia subgroup I Qp Gal(Q p /Q p ) that factors through the tame quotient) corresponding to τ via the inertial Langlands correspondence (c.f. Section 3 of [CEGGPS]). The notation LL -1 (τ ) comes from the compatibility between classical local Langlands correspondence and inertial local Langlands correspondence. Then we consider all lifts ρ of ρ with Hodge-Tate weights {0, 1,

• • • , n -1} such that WD(ρ)| I Qp ∼ = LL -1 (τ ). (2.2.4)
We note that each such ρ corresponds to a E-point of a formal scheme given by a complete Noetherian local ring R

LL -1 (τ ),{0,••• ,n-1} ρ
which is a special case of the construction of potentially semi-stable deformation rings in [START_REF] Kisin | Potentially semistable deformation rings[END_REF] for more general inertial types and Hodge-Tate weights.

For each pair of integers (i, j) such that 1 ≤ i < i + 1 < j ≤ n, we will choose a Deligne-Lusztig representation τ i,j and then pick a Galois representation r i,j : Gal(F /F ) → GL n (E) such that (i) r i,j is absolutely irreducible, conjugate self-dual and unramified at each finite place of F above a certain v ∈ D(U v0 );

(ii) r i,j contains a Galois-invariant O E -lattice r • i,j whose reduction modulo E is r;

(iii) ρ i,j := r i,j | Gal(Fw 0 /Fw 0 ) corresponds to a morphism R

LL -1 (τ ),{0,••• ,n-1} ρ → E.
In particular, we have the following commutative diagram

Π sm (r i,j ) Π(r i,j ) S(U v0 , O E )[p r • i,j ] S(U v0 , O E )[p r • i,j ] Π(r i,j ) τ i,j τ • i,j τ i,j ? O O _ ? o o ? O O _ ? o o / / ? O O ? O O _ ? o o ? O O / / / / (2.2.5)
where

τ • i,j := τ i,j ∩ S(U v0 , O E )[p r • i,j ] Π sm (r i,j
) and

τ i,j := τ • i,j ⊗ O E k E .
By taking I(1)-invariants, the diagram (2.2.5) induces another commutative diagram

Π sm (r i,j ) I(1) S(U v0 , O E )[p r • i,j ] I(1) Π(r i,j ) I(1) τ I(1) i,j (τ • i,j ) I(1) τ i,j I (1)
.

_ ? o o ? O O ? O O _ ? o o ? O O / / / / / / (2.2.6)
The fact that τ I(1) i,j = 0 implies that τ i,j is a principal series representation of GL n (F p ), which means that τ i,j comes from parabolic induction of a character of T (F p ). Each of the spaces in (2.2.6) carries an action of the Iwahori Hecke algebra which contains Ξ n and n -1 operators U m n ∈ Z p [GL n (Q p )] for 1 ≤ m ≤ n -1 which are usually called U p -operators. We use the notation P m for the standard parabolic of GL n containing the upper-triangular Borel and having Levi blocks of the form GL m × GL n-m . We denote the unipotent radical of P m by N m . The following lemma summarizes two main properties of U m n .

Lemma 2.2.7. We have

(Ξ n ) m • U m n ∈ Z p [GL n (Z p )]. Given a smooth irreducible representation Π m (resp. Π n-m ) of GL m (Q p ) (resp. of GL n-m (Q p )) with central character ω Πm (resp. ω Πn-m
). Then we have

U m n = ω Πm (p) -1
on the image of

Π m ⊗ E Π n-m → Ind GLn(Qp) Pm(Qp) (Π m ⊗ E Π n-m ) Nm(Zp)
.

We use the notation µ i,j for a character T (F p ) → Z × p (to be chosen) and µ i,j : T (F p ) → F × p for its reduction modulo p. We note that T (F p )-eigenspace Π sm (r i,j ) I(1),µi,j with eigencharacter µ i,j is also an eigenspace of U m n and we denote the corresponding eigenvalue (of U m n ) by α. Then we have the following picture capture α by applying an identity in Zp[GLn(Qp)] to (τ • i,j ) I(1),µ i,j τ I(1),µ i,j i,j

Π sm (ri,j ) I(1),µ i,j capture FLi,j (ρ) by applying an identity in Fp[GLn(Qp)] to τi,j I(1),µ i,j Π(ri,j ) I(1),µ i,j α α equals a product of eigenvalues of Frobenius of WD(ρi,j )

FL i,j (ρ) step 1 step 2 step 3 step 4 k s k s + 3
(2.2.8) which already appears in [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF]. We emphasize that m is an integer satisfying 1 ≤ m ≤ n -1 and is determined by the pair (i, j). The most standard part of picture 2.2.8 is step 3 which is an equality between the eigenvalue of U m n and a product of certain eigenvalues of Frobenius of WD(ρ i,j ) and follows directly from classical local Langlands correspondence and the classical local-global compatibility (c.f. Theorem 2.1.1). Then step 4 which realizes the invariant FL i,j (ρ) as reduction modulo E of a product of Frobenius eigenvalues (up to a factor which is a power of p), is proven using technical computations in integral p-adic Hodge theory, most notably Breuil and Kisin modules. The step 4 is mostly done in Section 3.3. We need to emphasize a significant difference between step 3 and 4. Step 3 lies completely in characteristic zero with coefficient E, depends only on WD(ρ) and requires no extra condition on ρ. On the other hand, step 4 relies on an extra technical condition on ρ which is going to be recalled in Definition 2.2.9.

Given a pair (ρ, τ i,j ), one can associate an element w(ρ, τ i,j ) in the affine Weyl group W of GL n via the theory of shapes essentially developed in [LLHLMa]. For each semi-simple Galois representation

ρ 0 : Gal(Q p /Q p ) → GL n (k E )
we define the following set Ω(ρ 0 , τ i,j ) := { w(ρ, τ i,j ) | ρ ss ∼ = ρ 0 }.

We consider the length ( w(ρ, τ i,j ))

with respect to the standard Coxeter system in W .

Definition 2.2.9. We say that ρ is τ i,j -generic if w(ρ, τ i,j ) has maximal length among all the elements in Ω(ρ ss , τ i,j ).

One can construct (via Fontaine-Laffaille modules) a k E -scheme M ρ 0 whose k E -points parameterize all Fontaine-Laffaille representation ρ with a fixed semi-simplification ρ ss ∼ = ρ 0 . Then we have a natural stratification S of M ρ 0 indexed by the set Ω(ρ 0 , τ i,j ) (this should follow from some work in preparation by authors of [LLHLMa] on moduli of Kisin modules with tamely ramified descent data). It turns out that M ρ 0 is irreducible and there is a unique element of maximal length in Ω(ρ 0 , τ i,j ) which corresponds to the unique (non-empty) open strata in S , which implies that τ i,j -generic is indeed a generic assumption. Concerning the terminology of [START_REF] Park | On mod p local global compatibility for GL n (Q p ) in the ordinary case[END_REF], the condition τ i,j -generic is essentially the same as saying that a subqutient of ρ determined by the pair (i, j) is Fontaine-Laffaille generic.

It follows from Lemma 2.2.7 that U m n = α i,j (2.2.10) on Π sm (r) I(1),µi,j for a certain constant α i,j ∈ E × and we set h i,j := val p (α i,j ), α i,j := p -hi,j α i,j ∈ O × E and let α i,j be the reduction modulo E of α i,j . The main output of step 3 and 4 in picture 2.2.8 is that Proposition 2.2.11. We have the equality

α i,j = FL i,j (ρ) -1 ∈ k × E (2.2.12) if ρ is τ i,j -generic.
It remains to clarify step 1 and 2 in picture 2.2.8. We need two group operators

S i,j , S i,j, ∈ Z p [GL n (F p )]
together with their reduction modulo p

S i,j , S i,j, ∈ F p [GL n (F p )]
which depends only on the choice of τ i,j and µ i,j . Then the identity in Z p [GL n (Q p )] that we need has the form

S i,j, • (Ξ n ) m • U m n = c i,j S i,j
(2.2.13) on Π sm (r) I(1),µi,j where c i,j ∈ E × is a constant depending only on the choice of τ i,j , µ i,j and m. We emphasize that (2.2.13) can be computed inside

Z p [GL n (Z p )] because (Ξ n ) m • U m n ∈ Z p [GL n (Z p )]
due to Lemma 2.2.7. Assume that p -hi,j c i,j ∈ O × E and let c i,j be the reduction modulo E of p -hi,j c i,j . Then we can combine (2.2.13) with (2.2.10) and obtain that

S i,j, • (Ξ n ) m = α -1 i,j c i,j S i,j (2.2.14)
on Π sm (r) I(1),µi,j whose reduction modulo E is

S i,j, • (Ξ n ) m = c i,j FL i,j (ρ)S i,j
(2.2.15) on Π(r) I(1),µi,j according to Proposition 2.2.11 and modulo an extra hypothesis:

Hypothesis 2.2.16. We have dim k E Π(r) I(1),µi,j = 1 and τ i,j I(1),µi,j ∼ -→ Π(r) I(1),µi,j .

One easily observes that Hypothesis 2.2.16 requires an optimal choice of the level U v0 , otherwise we need to replace it by another hypothesis claiming that a certain module over the Hecke algebra T is free over its support. To remove the Hypothesis 2.2.16, one needs a strong version of the condition τ i,j -generic again. In fact, it follows from a computation of Kisin modules with tamely ramified descent data (generalizing the computation in [LLHLMa] and will be proven in [LLMPQ]) that there exists a non-empty open subscheme

M sm ρ 0 of M ρ 0 contained in the τ i,j -generic open strata such that ρ is a k E -point of M sm ρ 0 if and only if R LL -1 (τi,j ),{0,••• ,n-1} ρ
is formally smooth. The very existence of M sm ρ 0 is an extremely strong restriction on the choice of τ i,j .

Definition 2.2.17. We say that ρ is strongly τ i,j -generic if it is given by a k E -point of M sm ρ 0 .

The following proposition (which is closely related to the Breuil-Mézard conjecture) will be proven in [LLMPQ] and essentially follows from a functor constructed by the celebrated Taylor-Wilies-Kisin patching method (c.f. Section 7.3 of [LLHLMa] for the case of GL 3 (Q p f )).

Proposition 2.2.18. The Hypothesis 2.2.16 is true for a suitable choice of U v0 if ρ is strongly τ i,jgeneric. Now we need to understand why do the group operators S i,j and S i,j, (satisfying (2.2.13)) exist and whether it is possible to explicitly construct them and compute the constant c i,j . The existence of certain S i,j and S i,j, that satisfy (2.2.13) is relatively formal and essentially follows from Hypothesis 2.2.16 but the group operators can be arbitrarily ugly and therefore we can say nothing about the constant c i,j . On the other hand, the explicit construction of a single pair of such group operators is very technical and the most difficult part is to show that S i,j and S i,j, are nonvanishing on Π(r) I(1),µi,j . In Chapter 3, we indeed (up to modifications on notation) give an explicit construction of group operators in Section 3.4.2, prove the nonvanishing property mentioned above in Section 3.4.8 and then compute the constant c i,j in Section 3.4.5.

In all, we conclude that the identity (2.2.15) captures the invariant FL i,j (ρ) inside Π(r) if ρ is strongly τ i,j -generic, and thus Π(r) determines ρ up to isomorphism if ρ is strongly τ i,j -generic for all pair (i, j) such that 1 ≤ i < i + 1 < j ≤ n.

There are various natural generalizations of Theorem 2.2.3 that have been largely done but not included in this thesis, and we will give a quick summary of them within three steps. We always assume in the following that ρ ss is sufficiently generic. The following generalizations are mostly work in preparation with Viet Bao Le Hung, Daniel Le, Stefano Morra and Chol Park.

The first step is to remove the condition ordinary. The new difficulty is divided into two parts: (i) one needs to choose a family of Fontaine-Laffaille invariants and a suitable τ for reach invariant such that the philosophy in Picture 2.2.8 works; (ii) one needs to explicitly construct certain pair of group operators S and S for each invariant such that an analogue of (2.2.13) is true with an explicit constant c. Note that we need to treat much more different cases of ρ (compared to ordinary case) and each case requires different choices of Fontaine-Laffaille invariants and of τ . The second step is to remove certain condition τ -generic which is analogous to Definition 2.2.9. The difficulty of this step is transparent as we do not expect R

LL -1 (τ ),{0,••• ,n-1} ρ
to be formally smooth without a strong generic assumption on ρ. Hence Proposition 2.2.11, Hypothesis 2.2.16 and (2.2.15) completely fail in general. The crucial observation is that we can determine ρ using a set of invariants together with a set of modular Serre weights. The last step is to remove the condition Fontaine-Laffaille. The problem is that ρ usually has lifts with respect to much fewer possible choices of τ if we fix Hodge-Tate weights. The natural solution is to to consider lifts (of ρ) with higher Hodge-Tate weights.

Remark 2.2.19. It should be possible to generalize all the discussions above to the case K = Q p f after a combination with recent work by Enns in [En]. However, if K is ramified over Q p , the situation seems to be much more complicated as Π(r) K(1) is usually too small and one may necessarily need wildly ramified inertial types to find sufficiently many invariants of ρ inside Π(r). The only known result with no restriction on the ramification of K is by Scholze in [START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] for n = 2 and arbitrary K using a completely different and more geometric method. It is not known (at least to the author) how to generalize the method in [START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] to arbitrary n.

Dilogarithm and higher

L -invariants for GL 3 (Q p )
Now we switch to the locally analytic representation Π an (r) which is the main focus of the second part of my thesis. One can apply the theory of Verma modules (namely the theory of Bernstein-Gelfand-Gelfand category O, c.f. [START_REF] Humphreys | Representations of Semisimple Lie Algebras in the BGG Category O[END_REF]) to study Π an (r). Very little is known about the full classification of irreducible objects in the category Rep la GLn(K),E consisting of admissible locally analytic representations of GL n (K). We consider the full subcategory Rep PS GLn(K),E (which is much better understood) of Rep la GLn(K),E generated by subquotients of locally analytic principal series representations Ind GLn(K) B(K) χ an for arbitrary locally analytic characters χ of T (K). When we say a full subcategory is generated by a family of objects, we mean that it is the minimal full subcategory (with each object of finite length) that contains this family of objects and is closed under extensions and taking subquotients. We assume that r is unramified at each finite place of F lying above D(U v0 ) and moreover Π alg (r) = 0.

(2.3.1)

Note that the condition (2.3.1) is the same as saying that r is automorphic and in particular ρ is potentially semi-stable. We define JH PS GLn(K),E

as the set of isomorphism classes of irreducible objects in Rep PS GLn(K),E and

JH PS GLn(K),E (r) ⊆ JH PS GLn(K),E
as the subset consisting of each irreducible object Π irr such that the embedding

Π irr → Π an (r)
exists. Moreover, we define the multiplicity

d Π irr := dim E Hom GLn(K) Π irr , Π an (r)
which is finite for each Π irr ∈ JH PS GLn(K),E (r) as Π an (r) is admissible. When ρ is crystalline and has generic Frobenius eigenvalues, the set JH PS GLn(K),E (r) is explicitly determined by certain combinatorial data given by the Hodge filtration of ρ due to [START_REF] Breuil | A local model for the trianguline variety and applications[END_REF], under certain standard technical assumption. On the other hand, [START_REF] Breuil | Towards the finite slope part for GL n , prépublication[END_REF] constructs a certain subrepresentation of Π an (r) which is maximal among those representations that satisfy a certain multiplicity one condition and belong to Rep PS GLn(K),E . The problem one might be interested in is to explicitly determine ρ using the structure of possibly a subrepresentation of Π an (r). If ρ is crystalline, evidences from Section 3.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and [START_REF] Breuil | Towards the finite slope part for GL n , prépublication[END_REF] suggest that it is still unclear how to construct a sufficiently large subrepresentation of Π an (r) to determine ρ completely, especially when the Hodge filtration of ρ lies in a sufficiently generic position. The situation is quite different (and actually much better) if ρ is semi-stable and non-crystalline, at least when n = 3. We assume from now on that F + = Q (hence K = Q p and v 0 = p) and that the (ϕ, N )-module D associated with WD(ρ) has the following form

D = Ee 0 ⊕ Ee 1 ⊕ Ee 2 (2.3.2)
where N (e 0 ) = 0, N (e 1 ) = e 0 , N (e 2 ) = e 1 and ϕ(e k ) = ur(α)p k-n+1 e k for 0 ≤ i ≤ 2 and a certain α ∈ E × . It follows from classical local Langlands correspondence that

LL • WD(ρ) = St ∞ 3 ⊗ E (ur(α) • det 3 )
where St ∞ 3 is the smooth Steinberg representation of GL 3 (Q p ) defined as the unique quotient of the smooth principal series Ind

GL3(Qp) B(Qp) 1 ∞ , det 3 is the determinant character GL 3 (Q p ) Q × p and ur(α) is the unramified character of Q × p which sends p to α. We use the notation k = (k 1 , k 2 , k 3 ) ∈ Z 3 with k 1 > k 2 > k 3
for the set of Hodge-Tate weights of ρ. We set

Π alg (k, D) := L(λ) ⊗ E St ∞ 3 ⊗ E (ur(α) • det 3 )
where L(λ) is the finite dimensional algebraic representation of GL 3 (Q p ) with highest weight

λ = (λ 1 , λ 2 , λ 3 ) = (k 1 -2, k 2 -1, k 3 ) (2.3.3)
with respect to the upper-triangular Borel subgroup B(Q p ). Then Theorem 2.1.1 implies that Π alg (r) = Π alg (k, D) ⊕d for an integer d = d Π alg (k,D) depending on U p and r. We say that ρ is non-critical (c.f. part (ii) of Remark 6.1.4 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]) if the Hodge filtration of ρ corresponds to a point in the open cell of a flag variety, which implies the following equality

JH PS GLn(K),E (r) = {Π alg (k, D)}. (2.3.4)
If ρ is non-criticial, then there exists a triple of invariants (L i,j ) 0≤i<j≤2 ∈ E 3 such that the (weakly admissible) Hodge filtration Fil • on D has the following form

Fil i D =        D if i ≤ k 3 E(e 2 + L 1,2 e 1 + L 0,2 e 0 ) ⊕ E(e 1 + L 0,1 e 0 ) if k 3 + 1 ≤ i ≤ k 2 E(e 2 + L 1,2 e 1 + L 0,2 e 0 ) if k 2 + 1 ≤ i ≤ k 1 0 if i ≥ k 1 + 1 .
(2.3.5)

One of the main purpose for pLL in the case (2.3.5) is to find all the invariants L i,j for 0 ≤ i < j ≤ 2 inside Π an (r). We usually use the terminology L -invariants for these L i,j and usually refer to simple L -invariants for L 0,1 and L 1,2 and call L 0,2 the higher L -invariant of ρ. What makes this case (2.3.5) so special is that one actually expects to find all the invariants L i,j inside a subrepresentation of Π an (r) that belongs to Rep PS GLn(K),E . The case for GL 2 (Q p ) was firstly studied by Breuil in [Bre04] and [START_REF] Breuil | Série spéciale p-adique et cohomologie étale complétée[END_REF] and was one of the starting points of pLL. In order to explicitly recover the unique L -invariant from Π an (r) if ρ is semi-stable non-crystalline of dimension two, one needs to consider the p-adic logarithm function to construct a basis for a certain Ext 1 -group. A recent result by Ding generalizes [START_REF] Breuil | Invariant L et série spéciale p-adique[END_REF] and [START_REF] Breuil | Série spéciale p-adique et cohomologie étale complétée[END_REF] to all the simple L -invariants (which are generalizations of L 0,1 and L 1,2 here) of GL n (K) for arbitrary K in [START_REF] Ding | Simple L-invariants for GL n[END_REF]. It remains to treat L 0,2 . In [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF], Schraen studied some locally analytic representations of GL 3 (Q p ) in [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] to try to realize the filtered module Fil • D inside the de Rham complex of the Drinfeld space with dimension two. One of the main technical ingredients in [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] is the computation of various (analytic) group cohomologies which leads to the determination of several crucial Ext-groups between certain locally analytic representations. Schraen constructs an abstract object Σ(λ, L ) (c.f. Definition 5.19 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]) in the derived category D Mod D(GL3(Qp),E) of the abelian category Mod D(GL3(Qp),E) consisting of abstract modules over the locally analytic distribution algebra D(GL 3 (Q p ), E). The motivation of this construction is to find a reasonable family of locally analytic representations of GL 3 (Q p ) that depends on three invariants, as ρ depends on three invariants. In fact, Schraen firstly constructs a family of locally analytic representations Σ(λ, L , L ) (c.f. (1.11) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]) depending on two invariants. Then he observes that a third invariant lies inside the group

Ext 2 GL3(Qp) (L(λ), Σ(λ, L , L )) (2.3.6)
rather than Ext 1 GL3(Qp) (L(λ), Σ(λ, L , L )), and thus one has to replace a family of locally analytic representations with a family of complex Σ(λ, L ) . Roughly speaking, one wishes to add L(λ) to obtain the third invariant but only succeeds in the sense of an abstract complex in [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. A crucial observation by Breuil is that one can possibly extend each Σ(λ, L , L ) to a larger locally analytic representation that contains more invariants by adding some constituents other than L(λ). Along this line, Breuil constructed in [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] two locally analytic representations Π 1 (k, D) and Π 2 (k, D) of GL 3 (Q p ) depending only on k and D such that

dim E Ext 1 GL3(Qp) Π i (k, D), Π alg (k, D) = 3 (2.3.7)
for i = 1, 2. To summary, we need the following technical conditions Condition 2.3.8. Assume that U p and r satisfy the following (i) Π alg (r) = 0;

(ii) WD(ρ) is given by the case n = 3 of (2.3.2);

(iii) ρ is non-critical;

(iv) the automorphic representation corresponding to r appears only once in the space of automorphic forms on G(A ∞ F + ). Then it is shown in Theorem 6.2.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] that for each fixed pair (U p , r) satisfying Condition 2.3.8, there exists a unique locally analytic representation Π of GL 3 (Q p ) of the form

Π alg (k, D) Π 1 (k, D) Π 2 (k, D) (2.3.9) such that Hom GL3(Qp) Π ⊗ E ε 2 • det 3 , Π an (r) ∼ -→ Hom GL3(Qp) Π alg (k, D) ⊗ E ε 2 • det 3 , Π an (r) . (2.3.10)
Note that we use the notation V W for a non-split extension of W by V . One crucial observation is that Π depends on four invariants according to (2.3.7) but ρ depends only on three invariants according to (2.3.5). In particular, we do not expect all locally analytic representations of the form (2.3.9) to be able to embed into Π an (r) for certain U p and r. A natural question is the following Question 2.3.11. Is there a purely representation theoretic criterion for a representation of the form (2.3.9) to embed into a certain Π an (r)?

The key point is that L(λ) is exactly the obstruction we want in Question 2.3.11. More precisely, whether a locally analytic representation Π is included in a larger representation with L(λ) lying in the cosocle (actually with multiplicity two) gives a criterion for Π to embed into the completed cohomology. One very interesting insight of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] is that the p-adic dilogarithm function appears in the two dimensional space (2.3.6) (c.f. Corollary 5.17 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]). Consequently, Breuil has suggested that there should be some explicit relation between [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF], [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] especially in the sense that the p-adic dilogarithm function should be related to the unique higher L -invariant L 0,2 of ρ which should be analogues to the relation between the p-adic logarithm function and simple L -invariants. We have arrived at the following question Question 2.3.12. What is the precise relation between the p-adic dilogarithm function and the unique higher L -invariant L 0,2 of ρ?

The main ingredient of the second part of my thesis is to try to answer the two questions at the same time through the construction of a family of locally analytic representations Σ min,+ (λ, L 1 , L 2 , L 3 ) depending on three invariants L 1 , L 2 , L 3 ∈ E (c.f. Theorem 4.1.1 for an intuitive description of the representation). The superscript 'min, +' means that Σ min,+ (λ, L 1 , L 2 , L 3 ) is slightly bigger than the minimal locally analytic representation of GL 3 (Q p ) which involves the p-adic dilogarithm function. Note that λ and k determine each other by the formula (2.3.3). The main properties of Σ min,+ (λ, L 1 , L 2 , L 3 ) are summarized in the following three results.

CHAPTER 2. GENERAL INTRODUCTION

Theorem 2.3.13. For each fixed U p and r satisfying Condition 2.3.8, there exists a unique choice of

L 1 , L 2 , L 3 ∈ E such that Hom GL3(Qp) Σ min,+ (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det 3 , Π an (r) ∼ -→ Hom GL3(Qp) Π alg (k, D) ⊗ E ε 2 • det 3 , Π an (r) . (2.3.14) Remark 2.3.15. We know that L 1 = L 1,2 , L 2 = L 0,1 (2.3.16)
due to our definition of L 1 and L 2 , Theorem 1.2 of [START_REF] Ding | Simple L-invariants for GL n[END_REF] and certain p-adic Hodge theoretic arguments appearing in the GL 2 (Q p ) case (c.f. Remark 3.1 of [START_REF] Ding | Simple L-invariants for GL n[END_REF]). The statement of Theorem 2.3.13 implies that L 3 is uniquely determined by U p and r, but a priori not necessarily by ρ. It is strongly believed but not yet known how to prove (using purely representation theoretic methods) that there is an embedding The following is a positive answer to Question 2.3.11 (at least in one direction) and easily follows from Theorem 6.2.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and our Theorem 2.3.13.

Π(D) → Σ min,+ (λ, L 1 , L 2 , L 3 ) (2.3.17) for certain L 1 , L 2 , L 3 ∈ E
Corollary 2.3.18. We fix a representation Π of the form (2.3.9). If we have an embedding

Π ⊗ E ε 2 • det 3 → Π an (r)
for a suitable choice of U p and r, then we have an embedding

Π → Σ min,+ (λ, L 1 , L 2 , L 3 ) ⊗ E ur(α) • det 3 for certain L 1 , L 2 , L 3 ∈ E uniquely determined by Π.
The following result is a natural consequence of the construction of Σ min,+ (λ, L 1 , L 2 , L 3 ), more details can be found in Theorem 4.1.7.

Theorem 2.3.19. There exists an explicit complex of locally analytic representations of GL 3 (Q p ) whose strong dual realizes the object Σ(λ, L ) (up to changes on notation, c.f. (1.14) and (5.19) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]).

Chapter 3

Mod p local-global compatibility for GL n (Q p ) in the ordinary case

Introduction

It is believed that one can attach a smooth F p -representation of GL n (K) (or a packet of such representations) to a continuous Galois representation Gal(Q p /K) → GL n (F p ) in a natural way, that is called mod p Langlands program for GL n (K), where K is a finite extension of Q p . This conjecture is well-understood for GL 2 (Q p ) ([BL94], [START_REF] Berger | Représentations modulaires de GL 2 (Q p ) et représentations galoisiennes de dimension 2[END_REF], [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) I[END_REF], [START_REF] Breuil | Sur quelques représentations modulaires et p-adiques de GL 2 (Q p ) II[END_REF], [START_REF] Colmez | Représentations de GL 2 (Q p ) et ( , Γ)-modules[END_REF], [START_REF] Paskunas | The image of Colmez's Montreal functor[END_REF], [CDP], [Eme]). Beyond the GL 2 (Q p )-case, for instance GL n (Q p ) for n > 2 or even GL 2 (Q p f ) for an unramified extension Q p f of Q p of degree f > 1, the situation is still quite far from being understood. One of the main difficulties is that there is no classification of such smooth representations of GL n (K) unless K = Q p and n = 2: in particular, we barely understand the supercuspidal representations. Some of the difficulties in classifying the supercuspidal representations are illustrated in [START_REF] Breuil | Towards a modulo p Langlands correspondence for GL 2 Memoirs of[END_REF], [START_REF] Hu | Sur quelques représentations supersingulières de GL 2 (Q p f )[END_REF] and [START_REF] Schraen | Sur la présentation des représentations supersingulières de GL 2 (F )[END_REF].

Let F be a CM field in which p is unramified, and r : Gal(Q/F ) → GL n (F p ) an automorphic Galois representation. Although there is no precise statement of mod p Langlands correspondence for GL n (K) unless K = Q p and n = 2, one can define smooth representations Π(r) of GL n (F w ) in the spaces of mod p automorphic forms on a definite unitary group cut out by the maximal ideal of a Hecke algebra associated to r, where w is a place of F above p. A precise definition of Π(r) when p splits completely in F , which is our context, will be given in Section 3.1.4. (See also Section 3.5.6.) One wishes that Π(r) is a candidate on the automorphic side corresponding to r| Gal(Q p /Fw) for a mod p Langlands correspondence in the spirit of Emerton [Eme]. However, we barely understand the structure of Π(r) as a representation of GL n (F w ), though the ordinary part of Π(r) is described in [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF] when p splits completely in F and r| Gal(Q p /Fw) is ordinary. In particular, it is not known whether Π(r) and r| Gal(Q p /Fw) determine each other. But we have the following conjecture:

Conjecture 3.1.1. The local Galois representation r| Gal(Q p /Fw) is determined by Π(r).
This conjecture is widely expected to be true by experts but not explicitly written down before. The case GL 2 (Q p f ) was treated by Breuil-Diamond [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF]. Herzig-Le-Morra [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] considered the case GL 3 (Q p ) when r| Gal(Q p /Fw) is upper-triangular, while the case GL 3 (Q p ) when r| Gal(Q p /Fw) is an extension of a two dimensional irreducible representation by a character was considered by Le-Morra-Park [LMP]. A recent work of John Enns studied this conjecture for the group GL 3 (Q p f ). All of the results above are under certain generic assumptions on the tamely ramified part of r| Gal(Q p /Fw) . a division algebra D over K with center K and invariant 1 n , which also has a continuous action of Gal(Q p /K), via the mod p cohomology of the Lubin-Tate tower. Using this construction, it was possible for Scholze to prove Conjecture 3.1.1 in full generality for GL 2 (K) (c.f. [Sch15], Theorem 1.5). On the other hand, the proof of Theorem 1.5 of [START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] does not tell us where the invariants that determine S(Π) lie. We do not know if there is any relation between these two different methods.

The weight part of Serre's conjecture already gives part of the information of Π(r): the local Serre weights of r at w determine the socle of Π(r)| GLn(O Fw ) at least up to possible multiplicities, where O Fw is the ring of integers of F w . If r| Gal(Q p /Fw) is semisimple, then it is believed that the Serre weights of r at w determine r| Gal(Q p /Fw) up to twisting by unramified characters, but this is no longer the case if it is not semisimple: the Serre weights are not enough to determine the wildly ramified part of r| Gal(Q p /Fw) , so that we need to understand a deeper structure of Π(r) than just its GL n (O Fw )-socle.

In this paper, we show that Conjecture 3.1.1 is true when p splits completely in F and r| Gal(Q p /Fw) is upper-triangular and sufficiently generic in a precise sense. Moreover, we describe the invariants in Π(r) that determine the wildly ramified part of r| Gal(Q p /Fw) . The generic assumptions on r| Gal(Q p /Fw) ensure that very few Serre weights of r at w will occur, which we call the weight elimination conjecture, Conjecture 3.1.11. The weight elimination results are significant for our method to prove Conjecture 3.1.1. But thanks to Bao V. Le Hung, this weight elimination conjecture is known to be true and will be proven in [LLMPQ]. We follow the basic strategy in [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF] and [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF]: we define Fontaine-Laffaille parameters on the Galois side using Fontaine-Laffaille modules as well as automorphic parameters on the automorphic side using the actions of Jacobi sum operators, and then identify them via the classical local Langlands correspondence. However, there are many new difficulties that didn't occur in [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF] or in [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF]. For instance, the classification of semi-linear algebraic objects of rank n > 3 on the Galois side is much more complicated. Moreover, failing of the multiplicity one property of the Jordan-Hölder factors of mod p reduction of Deligne-Lusztig representations of GL n (Z p ) for n > 3 implies that new ideas are required to show crucial non-vanishing of the automorphic parameters. In the rest of the introduction, we explain our ideas and results in more detail.

Local Galois side

Let E be a (sufficiently large) finite extension of Q p with ring of integers O E , a uniformizer E , and residue field F, and let I Qp be the inertia subgroup of Gal(Q p /Q p ) and ω the fundamental character of niveau 1. We also let ρ 0 : Gal(Q p /Q p ) → GL n (F) be a continuous (Fontaine-Laffaille) ordinary generic Galois representation. Namely, there exists a basis e := (e n-1 , e n-2 , • • • , e 0 ) for ρ 0 such that with respect to e the matrix form of ρ 0 is written as follows:

ρ 0 | I Qp ∼ =          ω cn-1+(n-1) * n-1 * • • • * * 0 ω cn-2+(n-2) * n-2 • • • * * 0 0 ω cn-3+(n-3) • • • * * . . . . . . . . . . . . . . . . . . 0 0 0 • • • ω c1+1 * 1 0 0 0 • • • 0 ω c0          (3.1.2)
for some integers c i satisfying some genericity conditions (c.f. Definition 3.3.3). We also assume that ρ 0 is maximally non-split, i.e.,

* i = 0 for all i ∈ {1, 2, • • • , n -1}.
Our goal on the Galois side is to show that the Frobenius eigenvalues of certain potentially crystalline lifts of ρ 0 determine the Fontaine-Laffaille parameters of ρ 0 , which parameterizes the wildly ramified part of ρ 0 . When the unramified part and the tamely ramified part of ρ 0 are fixed, we define the Fontaine-Laffaille parameters via the Fontaine-Laffaille modules corresponding to ρ 0 (c.f. Definition 3.3.15). These parameters vary over the space of (n-1)(n-2) and we write FL i0,j0 n (ρ 0 ) ∈ P 1 (F) for each pair of integers (i 0 , j 0 ) with 0 ≤ j 0 < j 0 +1 < i 0 ≤ n-1. For each such pair (i 0 , j 0 ), the Fontaine-Laffaille parameter FL i0,j0 n (ρ 0 ) is determined by the subquotient ρ i0,j0 of ρ 0 which is determined by the subset (e i0 , e i0-1 , • • • , e j0 ) of e (c.f. (3.3.2)): in fact, we have the identity FL i0,j0 n (ρ 0 ) = FL i0-j0,0 i0-j0+1 (ρ i0,j0 ) (c.f. Lemma 3.3.17). Since potentially crystalline lifts of ρ 0 are not Fontaine-Laffaille in general, we are no longer able to use Fontaine-Laffaille theory to study such lifts of ρ 0 ; we use Breuil modules and strongly divisible modules for their lifts. It is obvious that any lift of ρ 0 determines the Fontaine-Laffaille parameters, but it is not obvious how one can explicitly visualize the information that determines ρ 0 in those lifts. Motivated by the automorphic side, we believe that for each pair (i 0 , j 0 ) as above the Fontaine-Laffaille parameter FL i0,j0 n (ρ 0 ) is determined by a certain product of Frobenius eigenvalues of the potentially crystalline lifts of ρ 0 with Hodge-Tate weights

{-(n -1), • • • , -1, 0} and Galois type n-1 i=0 ω k i 0 ,j 0 i
where ω is the Teichmüler lift of the fundamental character ω of niveau 1 and

k i0,j0 i ≡    c i0 + i 0 -j 0 -1 for i = i 0 ; c j0 -(i 0 -j 0 -1) for i = j 0 ; c i otherwise (3.1.3) modulo (p -1).
Here, c i are the integers determining the tamely ramified part of ρ 0 in (3.1.2) and our normalization of the Hodge-Tate weight of the cyclotomic character ε is -1.

Our main result on the Galois side is the following:

Theorem 3.1.4 (Theorem 3.3.44). Fix i 0 , j 0 ∈ Z with 0 ≤ j 0 < j 0 +1 < i 0 ≤ n-1.
Assume that ρ 0 is generic (c.f. Definition 3.3.3) and that ρ i0,j0 is Fontaine-Laffaille generic (c.f. Definition 3.3.18), and

let (λ i0,j0 n-1 , λ i0,j0 n-2 , • • • , λ i0,j0 0 ) ∈ (O E ) n be the Frobenius eigenvalues on the ( ω k i 0 ,j 0 n-1 , ω k i 0 ,j 0 n-2 , • • • , ω k i 0 ,j 0 0 )- isotypic components of D Qp,n-1 st (ρ 0 ) where ρ 0 is a potentially crystalline lift of ρ 0 with Hodge-Tate weights {-(n -1), -(n -2), • • • , -1, 0} and Galois type n-1 i=0 ω k i 0 ,j 0 i .
Then the Fontaine-Laffaille parameter FL i0,j0 n associated to ρ 0 is computed as follows:

FL i0,j0 n (ρ 0 ) =   1 : p [(n-1)-i 0 +j 0 2 ](i0-j0-1) i0-1 i=j0+1 λ i0,j0 i   ∈ P 1 (F).
Note that by • ∈ F in the theorem above we mean the image of • ∈ O E under the natural surjection O E F. We also note that ρ i0,j0 being Fontaine-Laffaille generic implies FL i0,j0 n (ρ 0 ) = 0, ∞ for all i 0 , j 0 as in Theorem 3.1.4, but is a strictly stronger assumption if i 0 -j 0 ≥ 3.

Let us briefly discuss our strategy for the proof of Theorem 3.1.4. Recall that the Fontaine-Laffaille parameter FL i0,j0 n (ρ 0 ) is defined in terms of the Fontaine-Laffaille module corresponding to ρ 0 . Thus we need to describe FL i0,j0 n (ρ 0 ) by the data of the Breuil modules of inertial type n-1 i=0 ω k i 0 ,j 0 i corresponding to ρ 0 , and we do this via étale φ-modules, which requires classification of such Breuil modules. If the filtration of the Breuil modules is of a certain shape, then a certain product of the Frobenius eigenvalues of the Breuil modules determines a Fontaine-Laffaille parameter (c.f. Proposition 3.3.32). In order to get such a filtration, we need to assume that ρ i0,j0 is Fontaine-Laffaille generic (c.f. Definition 3.3.18). Then we determine the structure of the filtration of the strongly divisible modules lifting the Breuil modules by direct computation, which immediately gives enough properties of Frobenius eigenvalues of the potentially crystalline representations we consider. But this whole process is subtle for general i 0 , j 0 . To resolve this issue we prove that any potentially crystalline lift of ρ 0 with Hodge-Tate weights {-

(n-1), -(n-2), • • • , 0} and Galois type n-1 i=0 ω k i 0 ,j 0 i has a potentially crystalline subquotient ρ i0,j0 of Hodge-Tate weights {-i 0 , • • • , -j 0 } and of Galois type i0 i=j0 ω k i 0 ,j 0 i lifting ρ i0,j0 . More precisely, 32CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Theorem 3.1.5 (Corollary 3.3.41). Every potentially crystalline lift ρ 0 of ρ 0 with Hodge-Tate weights {-(n -1), -(n -2), • • • , 0} and Galois type n-1 i=0 ω k i 0 ,j 0 i is a successive extension ρ 0 ∼ =             ρ n-1,n-1 • • • * * * • • • * . . . . . . . . . . . . . . . . . . ρ i0+1,i0+1 * * • • • * ρ i0,j0 * • • • * ρ j0-1,j0-1 • • • * . . . . . . ρ 0,0            
where

• for n -1 ≥ i > i 0 and j 0 > i ≥ 0, ρ i,i is a 1-dimensional potentially crystalline lift of ρ i,i with
Hodge-Tate weight -i and Galois type ω k i 0 ,j 0 i ;

• ρ i0,j0 is a (i 0 -j 0 + 1)-dimensional potentially crystalline lift of ρ i0,j0 with Hodge-Tate weights

{-i 0 , -i 0 + 1, • • • , -j 0 } and Galois type i0 i=j0 ω k i 0 ,j 0 i .
Note that we actually prove the niveau f version of Theorem 3.1.5 since it adds only little more extra work (c.f. Corollary 3.3.41).

The representation ρ i0,j0 ⊗ ε -j0 is a (i 0 -j 0 + 1)-dimensional potentially crystalline lift of ρ i0,j0

with Hodge-Tate weights {-

(i 0 -j 0 ), -(i 0 -j 0 -1), • • • , 0} and Galois type i0 i=j0 ω k i 0 ,j 0 i
, so that, by Theorem 3.1.5, Theorem 3.1.4 reduces to the case (i 0 , j 0 ) = (n -1, 0): we prove Theorem 3.1.4 when (i 0 , j 0 ) = (n -1, 0), and then use the fact FL i0,j0 n

(ρ 0 ) = FL i0-j0,0 i0-j0+1 (ρ i0,j0
) to get the result for general i 0 , j 0 .

The Weil-Deligne representation WD(ρ 0 ) associated to ρ 0 (as in Theorem 3.1.4) contains those Frobenius eigenvalues of ρ 0 . We then use the classical local Langlands correspondence for GL n to transport the Frobenius eigenvalues of ρ 0 (and so the Fontaine-Laffaille parameters of ρ 0 as well by Theorem 3.1.4) to the automorphic side (c.f. Corollary 3.3.46).

Local automorphic side

We start by introducing the Jacobi sum operators in characteristic p. Let T (resp. B) be the maximal torus (resp. the maximal Borel subgroup) consisting of diagonal matrices (resp. of upper-triangular matrices) of GL n . We let X(T ) := Hom(T, G m ) be the group of characters of T and Φ + be the set of positive roots with respect to (B, T ). We define i ∈ X(T ) as the projection of T ∼ = G n m onto the i-th factor. Then the elements { i | 1 ≤ i ≤ n} forms a Z-basis for the free abelian group X(T ). We will use the notation

(d 1 , d 2 , • • • , d n ) ∈ Z n for the element n k=1 d k k ∈ X(T ).
Note that the group of characters of the finite group T (F p ) ∼ = (F × p ) n can be identified with X(T )/(p -1)X(T ), and therefore we sometimes abuse the notation (

d 1 , d 2 , • • • , d n ) for its image in X(T )/(p -1)X(T ). We define ∆ := {α k := k -k+1 | 1 ≤ k ≤ n -1} ⊂ Φ +
as the set of positive simple roots. Note that we write s k for the reflection of the simple root α k . For an element w in the Weyl group W , we define

Φ + w = {α ∈ Φ + | w(α) ∈ -Φ + } ⊆ Φ + and U w = α∈Φ + w U α
, where U α is a subgroup of U whose only non-zero off-diagonal entry corresponds to α. Note in particular that Φ + = Φ + w0 , where w 0 is the longest element in W . For w ∈ W and for a tuple of integers k

= (k α ) α∈Φ + w ∈ {0, 1, • • • , p -1} |Φ + w | , we define the Jacobi sum operator S k,w := A∈Uw(Fp)   α∈Φ + w A kα α   A • w ∈ F p [GL n (F p )] (3.1.6)
where A α is the entry of A corresponding to α ∈ Φ + w . In Section 3.4, we establish many technical results, both conceptual and computational, around these Jacobi sum operators. The use of these Jacobi sum operators can be traced back to at least [START_REF] Carter | Modular representations of finite groups of Lie type[END_REF], and are widely used for GL 2 in [START_REF] Breuil | Towards a modulo p Langlands correspondence for GL 2 Memoirs of[END_REF] and [START_REF] Hu | Sur quelques représentations supersingulières de GL 2 (Q p f )[END_REF] for instance. But systematic computation with these operators seems to be limited to GL 2 or GL 3 . In this paper, we need to do some specific but technical computation on some special Jacobi sum operators for GL n (F p ), which is enough for our application to Theorem 3.1.15 below.

By the discussion on the local Galois side, our target on the local automorphic side is to capture the Frobenius eigenvalues coming from the local Galois side. By the classical local Langlands correspondence, the Frobenius eigenvalues of ρ 0 are transported to the unramified part of χ in the tamely ramified principal series Ind GLn(Qp) B(Qp) χ corresponding to the Weil-Deligne representation WD(ρ 0 ) attached to ρ 0 in Theorem 3.1.4, and it is standard to use U p -operators to capture the information in the unramified part of χ.

The normalizer of the Iwahori subgroup I in GL n (Q p ) is cyclic modulo I, and this cyclic quotient group is generated by an element Ξ n ∈ GL n (Q p ) that is explicitly defined in (3.4.49). One of our goals is to translate the eigenvalue of U p -operators into the action of Ξ n on the space (Ind

GLn(Qp) B(Qp) χ)| GLn(Zp)
. This is firstly done for GL 2 (Q p f ) in [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF], and then the method is generalized to GL 3 (Q p ) in the ordinary case by [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF]. Both [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF] and [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] need a pair of group algebra operators: for instance, group algebra operators S, S ∈ Q p [GL 3 (Q p )] are defined in [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] and the authors prove an intertwining identity of the form S • Ξ 3 = c S on a certain I(1)-fixed subspace of Ind GL3(Qp) B(Qp) χ with χ assumed to be tamely ramified, where I(1) is the maximal pro-p subgroup of I. Here, the constant c ∈ O × E captures the eigenvalues of U p -operators. This is the first technical point on the local automorphic side, and we generalize the results in [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF] and [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] by the following theorem.

For an n-tuple of integers (

a n-1 , a n-2 , • • • , a 0 ) ∈ Z n , we write S n and S n for S k 1 ,w0 with k 1 = (k 1 i,j ) and S k 1, ,w0 with k 1, = (k 1, i,j ) respectively, where k 1 i,i+1 = [a 0 -a n-i ] 1 + n -2, k 1, i,i+1 = [a n-i-1 - a n-1 ] 1 +n-2 for 1 ≤ i ≤ n-1, and k 1 i,j = k 1, i,j = 0 otherwise. Here, (i, j) is the entry corresponding to α if α = i -j ∈ Φ + and by [x] 1 for x ∈ Z we mean the integer in [0, p -1) such that x ≡ [x] 1 modulo (p -1). We define S n ∈ Z p [GL n (Z p )] (resp. S n ∈ Z p [GL n (Z p )]
) by taking the Teichmüller lifts of the coefficients and the entries of the matrices of

S n ∈ F p [GL n (F p )] (resp. of S n ∈ F p [GL n (F p )]).
We use the notation • for the composition of maps or group operators to distinguish from the notation • for an O E -lattice inside a representation.

Theorem 3.1.7 (Theorem 3.4.71). Assume that the n-tuple of integers (a n-1 , a n-2 , • • • , a 0 ) is ngeneric in the lowest alcove (c.f. Definition 3.4.5), and let

Π n = Ind GLn(Qp) B(Qp) (χ 1 ⊗ χ 2 ⊗ χ 3 ⊗ ... ⊗ χ n-2 ⊗ χ n-1 ⊗ χ 0 )
be a tamely ramified principal series representation with the smooth characters χ k : 1),(a1,a2,...,an-1,a0) n we have the identity:

Q × p → E × satis- fying χ k | Z × p = ω a k for 0 ≤ k ≤ n -1. On the 1-dimensional subspace Π I(
S n • (Ξ n ) n-2 = p n-2 κ n n-2 k=1 χ k (p) S n (3.1.8) for κ n ∈ Z × p satisfying κ n ≡ ε * • P n (a n-1 , • • • , a 0 ) mod ( E ) where ε * = n-2 k=1 (-1) a0-a k and P n (a n-1 , • • • , a 0 ) = n-2 k=1 n-3 j=0 a k -a n-1 + j a 0 -a k + j ∈ Z × p .
In fact, there are many identities similar to the one in (3.1.8) for each operator U i n for 1 ≤ i ≤ n-1 (defined in (3.4.50)) which can be technically always reduced to Proposition 3.4.60, but it is clear from the proof of Theorem 3.1.7 in Section 3.4.5 that we need to choose U n-2 n for the U p -operator acting on Π I(1),(a1,a2,...,an-1,a0) n , motivated from the local Galois side via Theorem 3.1.4. The crucial point here is that the constant p n-2 κ n n-2 k=1 χ k (p) , which is closely related to FL n-1,0 n (ρ 0 ) via Theorem 3.1.4 and classical local Langlands correspondence, should lie in O × E for each Π n appearing in our application of Theorem 3.1.7 to Theorem 3.1.15.

The next step is to consider the mod p reduction of the identity (3.1.8), which is effective to capture 1),(a1,a2,...,an-1,a0) n

p n-2 n-2 k=1 χ k (p) modulo ( E ) only if S n v ≡ 0 modulo ( E ) for v ∈ Π I(
. It turns out that this non-vanishing property is very technical to prove for general GL n (Q p ). Before we state our non-vanishing result, we fix a little more notation: let

       µ * := (a n-1 -n + 2, a n-2 , • • • , a 1 , a 0 + n -2); µ 0 := (a n-1 , a 1 , • • • , a n-2 , a 0 ); µ 1 := (a 1 , a 2 , • • • , a n-3 , a n-2 , a n-1 , a 0 ); µ 1 := (a n-1 , a 0 , a 1 , a 2 , • • • , a n-3 , a n-2 )
be four characters of T (F p ), and write π 0 (resp. π • 0 ) for the characteristic p principal series (resp. the characteristic 0 principal series) induced by the characters µ 0 (resp. by its Teichmüller lift µ 0 ). Note that we can attach an irreducible representation F (λ) of GL n (F p ) to each λ ∈ X(T )/(p -1)X(T ) satisfying some regular conditions (c.f. the beginning of Section 3.4). If we assume that (a n-1 , • • • , a 0 ) ∈ Z n is n-generic in the lowest alcove, the characters µ * , µ 0 , µ 1 and µ 1 do satisfy the regular condition and thus we have four irreducible representations

F (µ * ), F (µ 0 ), F (µ 1 ) and F (µ 1 ) of GL n (F p ). There is a unique (up to homothety) O E -lattice τ in π • 0 ⊗ O E E such that soc GLn(Fp) (τ ⊗ O E F) = F (µ * ).
We are now ready to state the non-vanishing theorem.

Theorem 3.1.9 (Corollary 3.4.37). Assume that the n-tuple of integers (a n-1 , a n-2 , • • • , a 0 ) is 2ngeneric in the lowest alcove (c.f. Definition 3.4.5).

Then we have

S n (τ ⊗ O E F) U (Fp),µ1 = 0 and S n (τ ⊗ O E F) U (Fp),µ 1 = 0.
The definition of µ 1 , µ 1 , µ 0 and µ * is motivated by our application of Theorem 3.1.9 to Theorem 3.1.15 and is closely related to the Galois types we choose in Theorem 3.1.4. We emphasize that, technically speaking, it is crucial that F (µ * ) has multiplicity one in π 0 . The proof of Theorem 3.1.9 is technical and makes full use of the results in Sections 3.4.1, 3.4.7, and 3.4.8.

Weight elimination and automorphy of a Serre weight

The weight part of Serre's conjecture is considered as a first step towards mod p Langlands program, since it gives a description of the socles of Π(r)| GLn(Zp) up to possible multiplicities. Substantial progress has been made for the groups GL 2 (O K ), where O K is the ring of integers of a finite extension

K of Q p ([BDJ10], [Gee11], [GK14], [GLS14], [ GLS15
]). For groups in higher semisimple rank, we also have a detailed description. (See [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], [LMP], [START_REF] Morra | Serre weights for three-dimensional ordinay Galois representations[END_REF], [LLHLMa] for GL 3 ; [Her09], [START_REF] Gee | Companion forms for unitary and sympletic groups[END_REF], [BLGG], [START_REF] Le | Weight elimination in Serre type conjectures[END_REF], [GHS] for general n.)

Weight elimination results are significant for the proof of our main global application, Theorem 3.1.15. For the purpose of this introduction, we quickly review some notation. Let F + be the maximal totally real subfield of a CM field F , and assume that p splits completely in F . Fix a place w of F above p and set v := w| F + . We assume that r is automorphic: this means that there exist a totally definite unitary group G n defined over F + that is an outer form of GL n/F + and split at places above p, an integral model

G n of G n such that G n × O F + v is reductive if v is a finite place of F + that splits in F , a compact open subgroup U = G n (O F + v ) × U v ⊆ G n (O F + v ) × G n (A ∞,v F + )
that is sufficiently small and unramified above p, a Serre weight

V = v |p V v that is an irreducible smooth F p -representation of G n (O F + ,p
), and a maximal ideal m r associated to r in the Hecke algebra acting on the space S(U, V ) of mod p algebraic automorphic forms such that

S(U, V )[m r ] = 0.
(3.1.10)

We write W (r) for the set of Serre weights V satisfying (3.1.10) for some U , and W w (r) for the set of local Serre weights

V v , that is irreducible smooth representations of G n (O F + v ) ∼ = GL n (O Fw ) ∼ = GL n (Z p ), such that V v ⊗ ( v =v V v ) ∈ W (r) for an irreducible smooth representation v =v V v of v =v G n (O F + v
). The local Serre weights V v have an explicit description as representations of GL n (F p ): there exists a p-restricted

(i.e. 0 ≤ a i -a i-1 ≤ p -1 for all 1 ≤ i ≤ n -1) weight a := (a n-1 , a n-2 , • • • , a 0 ) ∈ X(T ) such that F (a) ∼ = V v where F (a)
is the irreducible socle of the dual Weyl module associated to a (c.f. Section 3.5.2 as well as the beginning of Section 3.4).

Assume that r| Gal(Q p /Fw) ∼ = ρ 0 , where ρ 0 is defined as in (3.1.2). We define certain characters µ and µ ,i1,j1 of T (F p ) and a principal series

π i1,j1 * = Ind GLn(Fp) B(Fp) (µ ,i1,j1 ) w0
at the beginning of Section 3.5.3. Our main conjecture for weight elimination is Conjecture 3.1.11 (Conjecture 3.5.16). Assume that ρ i0,j0 is Fontaine-Laffaille generic and that µ ,i1,j1 is 2n-generic. Then we have an inclusion

W w (r) ∩ JH((π i1,j1 * ) ∨ ) ⊆ {F (µ ) ∨ , F (µ ,i1,j1 ) ∨ }. (3.1.12)
We emphasize that the condition ρ i0,j0 is Fontaine-Laffaille generic is crucial in Conjecture 3.1.11. For example, if n = 4 and (i 0 , j 0 ) = (3, 0) and we assume merely FL 3,0 4 (ρ 0 ) = 0, ∞ (which is strictly weaker than Fontaine-Laffaille generic), then we expect that an extra Serre weight can possibly appear in W w (r) ∩ JH((π i1,j1 * ) ∨ ). The Conjecture 3.1.11 is motivated by the proof of Theorem 3.1.4 and the theory of shape in [LLHLMa]. The special case n = 3 of Conjecture 3.1.11 was firstly proven in [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] and can also be deduced from the computations of Galois deformation rings in [LLHLMa].

Remark 3.1.13. In an earlier version of this paper, we prove Conjecture 3.1.11 for n ≤ 5. But our method is rather elaborate to execute for general n. We are informed that Bao V. Le Hung can prove Conjecture 3.1.11 completely and we have decided to put it inside [LLMPQ]. Therefore, Conjecture 3.1.11 becomes a theorem based on the results in [LLMPQ].

Finally, we also show the automorphy of the Serre weight F (µ ) ∨ . In other words,

F (µ ) ∨ ∈ W w (r) ∩ JH((π i1,j1 * ) ∨ ). (3.1.14)
Showing the automorphy of a Serre weight, in general, is very subtle. But thanks to the work of [BLGG] we are able to show the automorphy of F (µ ) ∨ by checking the existence of certain potentially diagonalizable crystalline lifts of ρ 0 (c.f. Proposition 3.5.17).

Mod p local-global compatibility

We now state our main results on mod p local-global compatibility. As discussed at the beginning of this introduction, we prove that Π(r) determines the ordinary representation ρ 0 . Moreover, we also describe the invariants in Π(r) that determine the wildly ramified parts of ρ 0 . We first recall the definition of Π(r). Keep the notation of the previous sections, and write b i = -c n-1-i for all 0 ≤ i ≤ n -1, with c i as in (3.1.2). We fix a place w of F above p and write v := w| F + , and we let r : G F → GL n (F) be an irreducible automorphic representation, of a Serre weight

V ∼ = v V v (c.f. Section 3.1.3), with r| G Fw ∼ = ρ 0 . Let V := v =v V v and set S(U v , V ) := lim -→ S(U v • U v , V ) where the direct limit runs over compact open subgroups U v ⊆ G n (O F + v ). This space S(U v , V ) has a natural smooth action of G n (F + v ) ∼ = GL n (F w ) ∼ = GL n (Q p )
by right translation as well as an action of a Hecke algebra that commutes with the action of G n (F + v ). We define

Π(r) := S(U v , V )[m r ]
where m r is the maximal ideal of the Hecke algebra associated to r. In the spirit of [Eme], this is a candidate on the automorphic side for a mod p Langlands correspondence corresponding to ρ 0 . Note that the definition of Π(r) relies on U v and V as well as choice of a Hecke algebra, but we suppress them in the notation. Fix n -1 ≥ i 0 > j 0 + 1 > j 0 ≥ 0, and define i 1 and j 1 by the equation i 1 + i 0 = j 1 + j 0 = n -1. Note that the following Jacobi sum operators

S i1,j1 , S i1,j1, , S i1,j1 1 , S i1,j1, 1 ∈ F p [GL j1-i1+1 (F p )]
are defined at the beginning of Section 3.4.2. Now we can state the main results in this paper.

Theorem 3.1.15 (Theorem 3.5.44). Fix a pair of integers (i 0 , j 0 ) satisfying 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1, and let r :

G F → GL n (F) be an irreducible automorphic representation with r| G Fw ∼ = ρ 0 . Assume that • µ ,i1,j1 is 2n-generic; • ρ i0,j0 is Fontaine-Laffaille generic. Assume further that {F (µ ) ∨ } ⊆ W w (r) ∩ JH((π i1,j1 * ) ∨ ) ⊆ {F (µ ) ∨ , F (µ ,i1,j1 ) ∨ }. (3.1.16)
Then there exists a primitive vector (c.f. Definition 3.5.43) in Π(r) I(1),µ i 1 ,j 1 1 . Moreover, for each primitive vector v i1,j1 ∈ Π(r)

I(1),µ i 1 ,j 1 1 , we have S i1,j1 • S i1,j1 1 v i1,j1 = 0 and S i1,j1, • S i1,j1, 1 • (Ξ n ) j1-i1-1 v i1,j1 = ε i1,j1 P i1,j1 (b n-1 , • • • , b 0 ) • FL i0,j0 n (r| G Fw ) • S i1,j1 • S i1,j1 1 v i1,j1
where

ε i1,j1 = j1-1 k=i1+1 (-1) bi 1 -b k -j1+i1+1 and P i1,j1 (b n-1 , • • • , b 0 ) = j1-1 k=i1+1 j1-i1-1 j=1 b k -b j1 -j b i1 -b k -j ∈ Z × p .
Note that the conditions in (3.1.16) can be removed under some standard Taylor-Wiles conditions (c.f. Remark 3.1.13 and (3.1.14)).

Theorem 3.1.15 relies on the choice of a principal series type (the niveau 1 Galois type

n-1 i=0 ω k i 0 ,j 0 i
). But this choice is somehow the unique one that could possibly make our strategy of the proof of Theorem 3.1.15 work.

Be careful that we cannot apply Theorem 3.1.7 and Theorem 3.1.9 directly to our local globalcompatibility for general (i 1 , j 1 ). Instead, we need to generalize Theorem 3.1.9 (resp. Theorem 3.1.7) to Proposition 3.5.38 (resp. Proposition 3.5.25) .

Corollary 3.1.17. Keep the notation of Theorem 3.1.15 and assume that each assumption in Theorem 3.1.15 holds for all (i 0 , j 0 ) such that 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1. Assume further that a freeness assumption is true (c.f. Remark 3.5.47 and note that this assumption can be removed by results in [LLMPQ]).

Then the structure of Π(r) as a admissible smooth F-representation of GL n (Q p ) determines the Galois representation ρ 0 up to isomorphism.

Notation

Much of the notation introduced in this section will also be (or have already been) introduced in the text, but we try to collect together various definitions here for ease of reading.

We let E be a (sufficiently large) extension of Q p with ring of integers O E , a uniformizer E , and residue field F. We will use these rings E, O E , and F for the coefficients of our representations. We also let K be a finite extension of Q p with ring of integers O K , a uniformizer , and residue field k. Let W (k) be the ring of Witt vectors over k and write K 0 for W (k)[ 1 p ]. (K 0 is the maximal absolutely unramified subextension of K.) In this paper, we are interested only in the fields K that are tamely ramified extension of Q p , in which case we let e := [K :

K 0 ] = p f -1 where f = [k : F p ].
For a field F , we write G F for Gal(F /F ) where F is a separable closure of F . For instance, we are mainly interested in G Qp as well as G K0 in this paper. The choice of a uniformizer ∈ K provides us with a map:

ω : G Qp -→W (k) : g -→ g( )
whose reduction mod ( ) will be denoted as ω . This map factors through Gal(K/Q p ) and ω | G K 0 becomes a homomorphism. Note that the choice of the embedding σ 0 : k → F provides us with a fundamental character of niveau f , namely

ω f := σ 0 • ω | Gal(K/K0
) , and we fix the embedding in this paper.

For a ∈ k, we write a for its Teichmüler lift in W (k). We also use the notation a for a, in particular, in Section 3.4.5. When the notation for an element • in k is quite long, we prefer

• to •. For instance, if a, b, c, d ∈ k then we write (a -b)(a -c)(a -d)(b -c)(b -d) for (a -b)(a -c)(a -d)(b -c)(b -d).
Note that ω is the Teichmüler lift of ω .

We normalize the Hodge-Tate weight of the cyclotomic character ε to be -1. Our normalization on class field theory sends the geometric Frobenius to the uniformizers. If a ∈ F × or a ∈ O × E then we write U a for the unramified character sending the geometric Frobenius to a. We may regard a character of G Qp as a character of Q × p via our normalization of class field theory. As usual, we write S for the p-adic completion of

W (k)[u, u ie i! ] i∈N , and let S O E := S ⊗ Zp O E and S E := S O E ⊗ Zp Q p . We also let S F := S O E /( E , Fil p S O E ) ∼ = (k ⊗ Fp F)[u]/u ep . Choose a uniformizer of K and let E(u) ∈ W (k)[u]
be the monic minimal polynomial of . The group Gal(K/K 0 ) acts on S via the character ω , and we write (S O E ) ω m for the ω m -isotypical component of S for m ∈ Z. We define (S F ) ω m in a similar fashion. If O E or F are clear, we often omit them, i.e., we write S ω m and S ω m for (S O E ) ω m and (S F ) ω m respectively. In particular, S 0 := S ω 0 ∼ = (k ⊗ Fp F)[u e ]/u ep and

S 0 := S ω 0 = ∞ i=0 a i E(u) i i! | a i ∈ W (k) ⊗ Zp O E and a i → 0 p-adically . 38CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
The association a ⊗ b → (σ(a)b) σ gives rise to an isomorphism k ⊗ Fp F ∼ = σ:k →F F, and we write e σ for the idempotent element in k ⊗ Fp F that corresponds to the idempotent element in σ:k →F F whose only non-zero entry is 1 at the position of σ.

To lighten the notation, we often write G for GL n/Zp . (By G n , we mean an outer form of GL n defined in Section 3.5.1.) We let B be the Borel subgroup of G consisting of upper-triangular matrices of G, U the unipotent subgroup of B, and T the torus of diagonal matrices of GL n . We also write B - and U -for the opposite Borel of B and the unipotent subgroup of B -, respectively. Let Φ + denote the set of positive roots with respect to (B, T ), and ∆ = {α k } 1≤k≤n-1 the subset of positive simple roots. We also let W be the Weyl group of GL n , which is often considered as a subgroup of GL n , and let s k be the simple reflection corresponding to α k . We write w 0 for the longest Weyl element in W , and we hope that the reader is not confused with places w or w of F .

We often write K for GL n (Z p ) for brevity. (Note that we use K for a tamely ramified extension of Q p as well, and we hope that it does not confuse the reader.) We will often use the following three open compact subgroups of GL n (Z p ): if we let red : GL n (Z p )

GL n (F p ) be the natural mod p reduction map, then

K(1) := Ker(red) ⊂ I(1) := red -1 (U (F p )) ⊂ I := red -1 (B(F p )) ⊂ K.
If M is a free F-module with a smooth action of K, then T (F p ) acts on the pro p Iwahori fixed subspace M I(1) via I/I(1) ∼ = T (F p ). We write M I(1),µ for the eigenspace with respect to a character µ : T (F p ) → F × p . M I(1) decomposes as

M I(1) ∼ = M I(1),µ
as T (F p )-representations, where the direct sum runs over the characters µ of T (F p ). In the obvious similar fashion, we define M I(1),µ when M is a free O E -module or a free E-module.

By [m] f for a rational number m ∈ Z[ 1 p ] ⊂ Q we mean the unique integer in [0, e) congruent to m mod (e) via the natural surjection Z[ 1 p ]
Z/eZ. By y for y ∈ R we mean the floor function of y, i.e., the biggest integer less than or equal to y. For a set A, we write |A| for the cardinality of A. If V is a finite-dimensional F-representation of a group H, then we write soc H V and cosoc H V for the socle of V and the cosocle of V , respectively. If v is a non-zero vector in a free module over F (resp. over O E , resp. over E), then we write

F[v] (resp. O E [v], resp. E[v]
) for the F-line (resp. the O E -line, resp. the E-line) generated by v.

We write x for the image of x ∈ O E under the natural surjection O E F. We also have a natural surjection P 1 (O E ) P 1 (F) defined by letting [x : y] ∈ P 1 (F) be the image of [x : y] ∈ P 1 (O E ) where

[x : y] = [1 : ( y x )] if y x ∈ O E ; [( x y ) : 1] if x y ∈ O E .
We often write y x for [x : y] ∈ P 1 (F) if x = 0.

Integral p-adic Hodge theory

In this section, we do a quick review of some (integral) p-adic Hodge theory which will be needed later. We note that all of the results in this section are already known or easy generalization of known results. We closely follow [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF] as well as [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] in this section.

Filtered (φ, N )-modules with descent data

In this section, we review potentially semi-stable representations and their corresponding linear algebra objects, admissible filtered (φ, N )-modules with descent data.

Let K be a finite extension of Q p , and K 0 the maximal unramified subfield of K, so that

K 0 = W (k) ⊗ Zp Q p
where k is the residue field of K. We fix the uniformizer p ∈ Q p , so that we fix an embedding B st → B dR . We also let K be a subextension of K with K/K Galois, and write φ ∈ Gal(K 0 /Q p ) for the arithmetic Frobenius.

A p-adic Galois representation ρ :

G K → GL n (E) is potentially semi-stable if there is a finite extension L of K such that ρ| G L is semi-stable, i.e., rank L0⊗E D K st (V ) = dim E V , where V is an underlying vector space of ρ and D K st (V ) := (B st ⊗ Qp V ) G L . We often write D K st (ρ) for D K st (V ). If K is the Galois closure of L over K , then ρ| G K is semi-stable, provided that ρ| G L is semi-stable. Definition 3.2.1. A filtered (φ, N, K/K , E)-module of rank n is a free K 0 ⊗ E-module D of rank n together with • a φ ⊗ 1-automorphism φ on D; • a nilpotent K 0 ⊗ E-linear endomorphism N on D; • a decreasing filtration {Fil i D k } i∈Z on D K = K ⊗ K0 D consisting of K ⊗ Qp E-submodules of D K
, which is exhaustive and separated;

• a K 0 -semilinear, E-linear action of Gal(K/K ) which commutes with φ and N and preserves the filtration on D K .

We say that D is (weakly) admissible if the underlying filtered (φ, N, K/K, E)-module is weakly admissible in the sense of [START_REF] Fontaine | Représentations p-adiques semi-stables[END_REF]. The action of Gal(K/K ) on D is often called descent data action. If V is potentially semi-stable, then D K st (V ) is a typical example of an admissible filtered (φ, N, K/K , E)-module of rank n.

Theorem 3.2.2 ([CF], Theorem 4.3).

There is an equivalence of categories between the category of weakly admissible filtered (φ, N, K/K , E)-modules of rank n and the category of n-dimensional potentially semi-stable E-representations of G K that become semi-stable upon restriction to G K .

Note that Theorem 3.2.2 is proved in [CF] in the case K = K , and that [START_REF] Savitt | On a conjecture of Conrad, Diamond, and Taylor[END_REF] gives a generalization to the statement with non-trivial descent data.

If V is potentially semi-stable, then so is its dual

V ∨ . We define D * ,K st (V ) := D K st (V ∨ ). Then D * ,K
st gives an anti-equivalence of categories from the category of n-dimensional potentially semistable E-representations of G K that become semi-stable upon restriction to G K to the category of weakly admissible filtered (φ, N, K/K , E)-modules of rank n, with quasi-inverse

V * ,K st (D) := Hom φ,N (D, B st ) ∩ Hom Fil (D K , B dR ).
It will often be convenient to use covariant functors. We define an equivalence of categories: for each

r ∈ Z V K ,r st (D) := V * ,K st (D) ∨ ⊗ ε r . The functor D K ,r st defined by D K ,r st (V ) := D K st (V ⊗ ε -r ) is a quasi-inverse of V K ,r st .
For a given potentially semi-stable representation ρ : G K → GL n (E), one can attach a Weil-Deligne representation WD(ρ) to ρ, as in [START_REF] Conrad | Modularity of certain potentially Barsotti-Tate Galois representations[END_REF], Appendix B.1. We refer to WD(ρ)| I Qp as to the Galois type associated to ρ. Note that WD(ρ) is defined via the filtered (φ, N, K/K , N )-module D K st (ρ) and that WD(ρ)| I K ∼ = WD(ρ ⊗ ε r )| I K for all r ∈ Z. Finally, we say that a potentially semi-stable representation ρ is potentially crystalline if the monodromy operator N on D K st (ρ) is trivial.

Strongly divisible modules with descent data

In this section, we review strongly divisible modules that correspond to Galois stable lattices in potentially semi-stable representations. We keep the notation of Section 3.2.1 From now on, we assume that K/K is a tamely ramified Galois extension with ramification index e(K/K ). We fix a uniformizer ∈ K with e(K/K ) ∈ K . Let e be the absolute ramification index of K and

E(u) ∈ W (k)[u] the minimal polynomial of over K 0 . Let S be the p-adic completion of W (k)[u, u ie i! ] i∈N .
The ring S has additional structures:

• a continuous, φ-semilinear map φ : S → S with φ(u) = u p and φ(

u ie i! ) = u pie i! ; • a continuous, W (k)-linear derivation of S with N (u) = -u and N ( u ie i! ) = -ie u ie i! ;
• a decreasing filtration {Fil i S} i∈Z ≥0 of S given by letting Fil i S be the p-adic completion of the

ideal j≥i E(u) j j! S;
• a group action of Gal(K/K ) on S defined for each g ∈ Gal(K/K ) by the continuous ring isomorphism g : S → S with g(w i

u i i/e ! ) = g(w i )h i g u i i/e ! for w i ∈ W (k), where h g ∈ W (k) satisfies g( ) = h g .
Note that φ and N satisfies N φ = pφN and that ĝ(E(u)) = E(u) for all g ∈ Gal(K/K ) since we assume e(K/K ) ∈ K . We write φ i for 1 p i φ on Fil i S. For i ≤ p -1 we have φ(Fil i S) ⊆ p i S.

Let S O E := S ⊗ Zp O E and S E := S O E ⊗ Zp Q p .
We extend the definitions of φ, N , Fil i S, and the action of Gal(K/K ) to S O E (resp. to S E ) O E -linearly (resp. E-linearly).

Definition 3.2.3. Fix a positive integer r < p -1. A strongly divisible O E -module with descent data of weight r is a free S O E -module M of finite rank together with

• a S O E -submodule Fil r M; • additive maps φ, N : M → M; • S O E -semilinear bijections g : M → M for each g ∈ Gal(K/K ) such that • Fil r S O E • M ⊆ Fil r M; • Fil r M ∩ I M = IFil r M for all ideals I in O E ;
• φ(sx) = φ(s)φ(x) for all s ∈ S O E and for all x ∈ M;

• φ(Fil r M) is contained in p r M and generates it over S O E ;

• N (sx) = N (s)x + sN (x) for all s ∈ S O E and for all x ∈ M;

• N φ = pφN ; • E(u)N (Fil r M) ⊆ Fil r M;
• for all g ∈ Gal(K/K ) g commutes with φ and N , and preserves Fil r M;

• g 1 • g 2 = g 1 • g 2 for all g 1 , g 2 ∈ Gal(K/K ).
We write O E -Mod r dd for the category of strongly divisible O E -modules with descent data of weight r. It is easy to see that the map

φ r = 1 p r φ : Fil r M → M satisfies cN φ r (x) = φ r (E(u)N (x)) for all x ∈ Fil r M where c := φ(E(u)) p ∈ S × .
For a strongly divisible O E -module M with descent data of weight r, we define a G K -module T * ,K st ( M) as follows (c.f. [EGH13], Section 3.1.):

T * ,K st ( M) := Hom Fil r ,φ,N ( M, A st ).
Proposition 3.2.4 ([EGH13], Proposition 3.1.4). The functor T * ,K st provides an anti-equivalence of categories from the category O E -Mod r dd to the category of G K -stable O E -lattices in finite-dimensional E-representations of G K which become semi-stable over K with Hodge-Tate weights lying in [-r, 0], when 0 < r < p -1.

Note that the case K = K and E = Q p is proved by Liu [START_REF] Liu | On lattices in semi-stable representations:a proof of a conjecture of Breuil[END_REF]. In this paper, we will be mainly interested in covariant functors T K ,r st from the category O E -Mod r dd to the category Rep

K-st,[-r,0] O E G K of G K -stable O E -lattices in finite-dimensional E-representations of G K which become semi-stable over K with Hodge-Tate weights lying in [-r, 0] defined by T K ,r st ( M) := T * ,K st ( M) ∨ ⊗ ε r .
Let M in O E -Mod r dd , and define a free S E -module D := M ⊗ Zp Q p . We extend φ and N on D, and define a filtration on D as follows: Fil r D = Fil r M[ 1 p ] and

Fil i D :=    D if i ≤ 0; {x ∈ D | E(u) r-i x ∈ Fil r D} if 0 ≤ i ≤ r; i-1 j=0 (Fil i-j S Qp )(Fil j D) if i > r, inductively.
(3.2.5)

We let D := D ⊗ S Qp ,s0 K 0 and D K := D ⊗ S Qp ,s K, where s 0 : S Qp → K 0 and s : S Qp → K are defined by u → 0 and u → respectively, which induce φ and N on D and the filtration on D K by taking s (Fil i D). The K 0 -vector space D also inherits an E-linear action and a semi-linear action of Gal(K/K ). Then it turns out that D is a weakly admissible filtered (φ, N, K/K , E)-module with Fil r+1 D = 0. Moreover, there is a compatibility (c.f.

[EGH13], Proof of Proposition 3.1.4.): if D corresponds to D = M[ 1 p ], then T K ,r st ( M)[ 1 p ] ∼ = V K ,r st (D).

Breuil modules with descent data

In this section, we review Breuil modules with descent data. We keep the notation of Section 3.2.2, and assume further that K ⊆ K 0 . We let

S := S/( E , Fil p S) ∼ = (k ⊗ Fp F)[u]/u ep .
It is easy to check that S inherits φ, N , the filtration of S, and the action of Gal(K/K ). Definition 3.2.6. Fix a positive integer r < p -1. A Breuil modules with descent data of weight r is a free S-module M of finite rank together with

• a S-submodule Fil r M of M;
• maps φ r : Fil r M → M and N : M → M;

• additive bijections g : M → M for all g ∈ Gal(K/K ) such that

• Fil r M contains u er M;
• φ r is F-linear and φ-semilinear (where φ : k[u]/u ep → k[u]/u ep is the p-th power map) with image generating M as S-module;

• N is k ⊗ Fp F-linear and satisfies

-N (ux) = uN (x) -ux for all x ∈ M,
u e N (Fil r M) ⊆ Fil r M, and

-φ r (u e N (x)) = cN (φ r (x)) for all x ∈ Fil r M, where c ∈ (k[u]/u ep ) × is the image of 1 p φ(E(u)) under the natural map S → k[u]/u ep .
• g preserves Fil r M and commutes with the φ r and N , and the action satisfies

g 1 • g 2 = g 1 • g 2 for all g 1 , g 2 ∈ Gal(K/K ). Furthermore, if a ∈ k ⊗ Fp F and m ∈ M then g(au i m) = g(a)(( g( ) ) i ⊗ 1)u i g(m).
We write F-BrMod r dd for the category of Breuil modules with descent data of weight r. For M ∈ F-BrMod r dd , we define a G K -module as follows (c.f. [EGH13], Section 3.2):

T * st (M) := Hom BrMod (M, A).
This gives an exact faithful contravariant functor from the category F-BrMod r dd to the category Rep F G K of finite dimensional F-representations of G K . We also define a covariant functor as follows: for each r ∈ Z T r st (M) := T * st (M) ∨ ⊗ ω r , in which we will be more interested in this paper.

If M is a strongly divisible module with descent data, then

M := M/( E , Fil p S)
is naturally an object in F-BrMod r dd (Fil r M is the image of Fil r M in M, the map φ r is induced by 1 p r φ| Fil r M , and N and g are those coming from M). Moreover, there is a compatibility: if

M ∈ O E -Mod r dd and we let M = M/( E , Fil p S) then T K ,r st ( M) ⊗ O E F ∼ = T r st (M).
(See [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Lemma 3.2.2 for detail.) There is a notion of duality of Breuil modules, which will be convenient for our computation of Breuil modules as we will see later.

Definition 3.2.7. Let M ∈ F-BrMod r dd . We define M * as follows:

• M * := Hom k[u]/u ep -Mod (M, k[u]/u ep ); • Fil r M * := {f ∈ M * | f (Fil r M) ⊆ u er k[u]/u ep }; • φ r (f ) is defined by φ r (f )(φ r (x)) = φ r (f (x)
) for all x ∈ Fil r M and f ∈ Fil r M * , where φ r :

u er k[u]/u ep → k[u]/u ep is the unique semilinear map sending u er to c r ; • N (f ) := N • f -f • N , where N : k[u]/u ep → k[u]/u ep is the unique k-linear derivation such that N (u) = -u; • ( gf )(x) = g(f ( g -1 x)) for all x ∈ M and g ∈ Gal(K/K ), where Gal(K/K ) acts on k[u]/u ep by g(au i ) = g(a)( g( ) ) i u i for a ∈ k.
If M is an object of F-BrMod r dd then so is M * . Moreover, we have M ∼ = M * * and

T * st (M * ) ∼ = T r st (M).
(c.f. [START_REF] Caruso | F p -représentations semi-stables[END_REF]), Section 2.1.) Finally, we review the notion of Breuil submodules developed mainly by [START_REF] Caruso | F p -représentations semi-stables[END_REF]. See also [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], Section 2.3.

Definition 3.2.8. Let M be an object of F-BrMod r dd . A Breuil submodule of M is an S-submodule N of M if N satisfies • N is a k[u]/u ep -direct summand of M; • N (N ) ⊆ N and g(N ) ⊆ N for all g ∈ Gal(K/K ); • φ r (N ∩ Fil r M) ⊆ N .
If N is a Breuil submodule of M, then N and M/N are also objects of F-BrMod r dd . We now state a crucial result we will use later.

Proposition 3.2.9 ([HLM17], Proposition 2.3.5). Let M be an object in F-BrMod r dd . Then there is a natural inclusion preserving bijection

Θ : {Breuil submodules in M} → {G K -subrepresentations of T r st (M)} sending N ⊆ M to the image of T r st (N ) → T r st (M). Moreover, if M 2 ⊆ M 1 are Breuil submodules of M, then Θ(M 1 )/Θ(M 2 ) ∼ = T r st (M 1 /M 2 ).
We will also need classification of Breuil modules of rank 1 as follows. We denote the Breuil modules in the following lemma by M(a, s, λ). Lemma 3.2.10 ([MP17], Lemma 3.1). Let k := F p f , e := p f -1, := e √ -p, and K = Q p . We also let M be a rank-one object in F-BrMod r dd . Then there exists a generator m ∈ M such that:

(i) M = S F • m; (ii) Fil r M = u s(p-1) M where 0 ≤ s ≤ re p-1 ; (iii) ϕ r (u s(p-1) m) = λm for some λ ∈ (F p f ⊗ Fp F) × ; (iv) g(m) = (ω f (g) a ⊗ 1)m for all g ∈ Gal(K/K 0 )
where a is an integer such that a + ps ≡ 0 mod ( e p-1 );

(v) N (m) = 0.
Moreover, one has

T r st (M)| I Qp = ω a+ps f .
The following lemma will be used to determine if the Breuil modules violate the maximal nonsplitness.

Lemma 3.2.11 ([MP17], Lemma 3.2). Let k := F p f , e := p f -1, := e √ -p, and K = Q p . We also let M x := M(k x , s x , λ x ) and M y := M(k y , s y , λ y ) be rank-one objects in F-BrMod r dd . Assume that the integers k x , k y , s x , s y ∈ Z satisfy

p(s y -s x ) + [k y -k x ] f > 0.
(3.2.12)

Assume further that f < p and let

0 → M x → M → M y → 0 be an extension in F-BrMod r dd , with T * st (M) being Fontaine-Laffaille. If the exact sequence of S F -modules 0 → Fil r M x → Fil r M → Fil r M y → 0 (3.2.13)
splits, then the G Qp -representation T * st (M) splits as a direct sum of two characters. In particular, provided that pk y ≡ k x modulo e and that s y (p -1) < re if f > 1, the representation T * st (M) splits as a direct sum of two characters if the element j 0 ∈ Z uniquely defined by

j 0 e + [p -1 k y -k x ] f < s x (p -1) ≤ (j 0 + 1)e + [p -1 k y -k x ] f (3.2.14) satisfies (r + j 0 )e + [p -1 k y -k x ] f < (s x + s y )(p -1). (3.2.15)

Linear algebra with descent data

In this section, we introduce the notion of framed basis for a Breuil module M and framed system of generators for Fil r M. Throughout this section, we assume that K 0 = K and continue to assume that K is a tamely ramified Galois extension of K . We also fix a positive integer r < p -1.

Definition 3.2.16. Let n ∈ N and let

(k n-1 , k n-2 , . . . , k 0 ) ∈ Z n be an n-tuple. A rank n Breuil module M ∈ F-BrMod r dd is of (inertial) type ω kn-1 ⊕ • • • ⊕ ω k0 if M has an S-basis (e n-1 , • • • , e 0 )
such that ge i = (ω ki (g) ⊗ 1)e i for all i and all g ∈ Gal(K/K 0 ). We call such a basis a framed basis of M.

We also say that f := (f n-1 , f n-2 , . . . , f 0 ) is a framed system of generators of Fil r M if f is a system of S-generators for Fil r M and gf i = (ω p -1 ki (g) ⊗ 1)f i for all i and all g ∈ Gal(K/K 0 ).

The existence of a framed basis and a framed system of generators for a given Breuil module M ∈ F-BrMod r dd is proved in [HLM17], Section 2.2.2. Let M ∈ F-BrMod r dd be of inertial type n-1 i=0 ω ki , and let e := (e n-1 , . . . , e 0 ) be a framed basis for M and f := (f n-1 , . . . , f 0 ) be a framed system of generators for Fil r M. The matrix of the filtration, with respect to e, f , is the matrix Mat e,f (Fil r M) ∈ M n (S) such that

f = e • Mat e,f (Fil r M).
Similarly, we define the matrix of the Frobenius with respect to e, f as the matrix Mat e,f (ϕ r ) ∈ GL n (S) characterized by

(φ r (f n-1 ), • • • , φ r (f 0 )) = e • Mat e,f (ϕ r ).
As we require e, f to be compatible with the framing, the entries in the matrix of the filtration satisfy the important additional properties:

Mat e,f (Fil r M) i,j ∈ S ω p f -1 k j -k i . More precisely, Mat e,f (Fil r M) i,j = u [p f -1 kj -ki] f s i,j , where s i,j ∈ S ω 0 = k ⊗ Fp F[u e ]/(u ep ).
We can therefore introduce the subspace M 2 n (S) of matrices with framed type τ

= n-1 i=0 ω ki f as M 2 n (S) := V ∈ M n (S) | V i,j ∈ S ω k j -k i f for all 0 ≤ i, j ≤ n -1 .
Similarly, we define

M 2, n (S) := V ∈ M n (S) | V i,j ∈ S ω p -1 k j -k i f for all 0 ≤ i, j ≤ n -1 and M 2, n (S) := V ∈ M n (S) | V i,j ∈ S ω p -1 (k j -k i ) f for all 0 ≤ i, j ≤ n -1 .
We also define GL

• n (S) := GL n (S) ∩ M • n (S) for • ∈ { } ∪ { , } ∪ { , }. As ϕ r (f i ) is a ω ki f -eigenvector for the action of Gal(K/K 0 ) we deduce that Mat e,f (Fil r M) ∈ M 2, n (S) and Mat e,f (ϕ r ) ∈ GL 2 n (S). Note that M 2 n (S) = M 2, n (S) = M 2, n (
S) if the framed type τ is of niveau 1. We use similar terminologies for strongly divisible modules M ∈ O E -Mod r dd . Definition 3.2.17. Let n ∈ N and let

(k n-1 , k n-2 , . . . , k 0 ) ∈ Z n be an n-tuple. A rank n strongly divisible module M ∈ O E -Mod r dd is of (inertial) type ω kn-1 ⊕ • • • ⊕ ω k0 if M has an S O E -basis e := ( e n-1 , • • • , e 0 )
such that g e i = ( ω ki (g) ⊗ 1) e i for all i and all g ∈ Gal(K/K 0 ). We call such a basis a framed basis for M. We also say that f := ( f n-1 , f n-2 , . . . , f 0 ) is a framed system of generators for Fil r M if f is a system of S-generators for Fil r M/Fil r S • M and g f i = ( ω p -1 ki (g) ⊗ 1) f i for all i and all g ∈ Gal(K/K 0 ).

One can readily check the existence of a framed basis for M and a framed system of generators for Fil r M, by Nakayama Lemma. We also define Mat e, f (Fil r M) and Mat e, f (φ r ) each of whose entries satisfies

Mat e, f (Fil r M) i,j ∈ S ω p f -1 k j -k i and Mat e, f (φ r ) i,j ∈ S ω k j -k i ,
in the similar fashion to Breuil modules. In particular,

Mat e, f (Fil r M) ∈ M 2, n (S) and Mat e, f (ϕ r ) ∈ GL 2 n (S)
where M , n (S) and GL n (S) are defined in the similar way to Breuil modules. We also define GL , n (S) in the similar way to Breuil modules again.

The inertial types on a Breuil module M and on a strongly divisible modules are closely related to the Weil-Deligne representation associated to a potentially crystalline lift of T r st (M).

Proposition 3.2.18 ([LMP], Proposition 2.12). Let M be an object in O E -Mod r dd and let M := M ⊗ S S/( E , Fil p S) be the Breuil module corresponding to the mod p reduction of M.

If T K0,r st ( M)[ 1 p ] has Galois type n-1 i=0 ω ki f for some integers k i , then M (resp. M) is of inertial type n-1 i=0 ω ki (resp. n-1 i=0 ω ki ). 46CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
Finally, we need a technical result on change of basis of Breuil modules with descent data. Lemma 3.2.19 ([HLM17], Lemma 2.2.8). Let M ∈ F-BrMod r dd be of type n-1 i=0 ω ki , and let e, f be a framed basis for M and a framed system of generators for Fil r M respectively. Write V := Mat e,f (Fil r M) ∈ M , n (S) and A := Mat e,f (ϕ r ) ∈ GL n (S), and assume that there are invertible matrices R ∈ GL n (S) and C ∈ GL , n (S) such that

R • V • C ≡ V mod (u e(r+1) ),
for some V ∈ M , n (S). Then e := e • R -1 forms another framed basis for M and f := e • V forms another framed system of generators for Fil r M such that

Mat e ,f (Fil r M) = V ∈ M , n (S) and Mat e ,f (φ r ) = R • A • φ(C) ∈ GL n (S).
In particular, if

R -1 = A then Mat e ,f (φ r ) = φ(C).
The statement of Lemma 3.2.19 is slightly more general than [HLM17], Lemma 2.2.8, but exactly the same argument works.

Fontaine-Laffaille modules

In this section, we briefly recall the theory of Fontaine-Laffaille modules over F, and we continue to assume that K 0 = K and that K is a tamely ramified Galois extension of K .

Definition 3.2.20. A Fontaine-Laffaille module over k ⊗ Fp F is the datum (M, Fil • M, φ • ) of • a free k ⊗ Fp F-module M of finite rank;
• a decreasing, exhaustive and separated filtration {Fil j M } j∈Z on M by k ⊗ Fp F-submodules;

• a φ-semilinear isomorphism φ • : gr • M → M , where gr • M := j∈Z Fil j M Fil j+1 M .
We write F-FLMod k for the category of Fontaine-Laffaille modules over k ⊗ Fp F, which is abelian. If the field k is clear from the context, we simply write F-FLMod to lighten the notation.

Given a Fontaine-Laffaille module M , the set of its Hodge-Tate weights in the direction of σ ∈ Gal(k/F p ) is defined as HT σ := i ∈ N | e σ Fil i M = e σ Fil i+1 M . In the remainder of this paper we will be focused on Fontaine-Laffaille modules with parallel Hodge-Tate weights, i.e. we will assume that for all i ∈ N, the submodules Fil i M are free over k ⊗ Fp F. Definition 3.2.21. Let M be a Fontaine-Laffaille module with parallel Hodge-Tate weights. A k⊗ Fp F basis f = (f 0 , f 1 , . . . , f n-1 ) on M is compatible with the filtration if for all i ∈ Z ≥0 there exists

j i ∈ Z ≥0 such that Fil i M = n j=ji k ⊗ Fp F • f j .
In particular, the principal symbols (gr(f 0 ), . . . , gr(f n-1 )) provide a k ⊗ Fp F basis for gr • M .

Note that if the graded pieces of the Hodge filtration have rank at most one then any two compatible basis on M are related by a lower-triangular matrix in GL n (k ⊗ Fp F). Given a Fontaine-Laffaille module and a compatible basis f , it is convenient to describe the Frobenius action via a matrix Mat f (φ • ) ∈ GL n (k ⊗ Fp F), defined in the obvious way using the principal symbols (gr(f 0 ), . . . , gr(f n-1 )) as a basis on gr • M .

It is customary to write F-FLMod [0,p-2] to denote the full subcategory of F-FLMod formed by those modules M verifying Fil 0 M = M and Fil p-1 M = 0 (it is again an abelian category). We have the following description of mod p Galois representations of G K0 via Fontaine-Laffaille modules: Proposition 3.2.22 ([FL82], Theorem 6.1). There is an exact fully faithful contravariant functor

T * cris,K0 : F-FLMod [0,p-2] k → Rep F (G K0 )
which is moreover compatible with the restriction over unramified extensions: if L 0 /K 0 is unramified with residue field l/k and if M is an object in F-FLMod

[0,p-2] k , then l ⊗ k M is naturally regarded as an object in F-FLMod [0,p-2] l and T * cris,L0 (l ⊗ k M ) ∼ = T * cris,K0 (M )| G L 0 .
We will often write T * cris for T * cris,K0 if the base field K 0 is clear from the context.

Definition 3.2.23. We say that ρ ∈ Rep F G K0 is Fontaine-Laffaille if T * cris (M ) ∼ = ρ for some M ∈ F-FLMod [0,p-2] .

Étale φ-modules

In this section, we review the theory of étale φ-modules, first introduced by Fontaine [Fon90], and its connection with Breuil modules and Fontaine-Laffaille modules. Throughout this section, we continue to assume that K 0 = K and that K is a tamely ramified Galois extension of K .

Let p 0 := -p, and let p be identified with a sequence (p n ) n ∈ Q p N verifying p p n = p n-1 for all n. We also fix := e √ -p ∈ K, and let 0 = . We fix a sequence ( n ) n ∈ Q p N such that e n = p n and p n = n-1 for all n ∈ N, and which is compatible with the norm maps

K( n+1 ) → K( n ) (c.f. [Bre14], Appendix A). By letting K ∞ := ∪ n∈N K( n ) and (K 0 ) ∞ := ∪ n∈N K 0 (p n ), we have a canonical isomorphism Gal(K ∞ /(K 0 ) ∞ ) ∼ -→ Gal(K/K 0 )
and we will identify ω as a character of Gal(K ∞ /(K 0 ) ∞ ). The field of norms k(( )) associated to (K, ) is then endowed with a residual action of Gal(K ∞ /(K 0 ) ∞ ), which is completely determined by g( ) = ω (g) .

We define the category φ, F ⊗ Fp k((p)) -Mod of étale (φ, F ⊗ Fp k((p)))-modules as the category of free F⊗ Fp k((p))-modules of finite rank M endowed with a semilinear map φ : M → M with respect to the Frobenius on k((p)) and inducing an isomorphism φ * M → M (with obvious morphisms between objects). We also define the category (φ, F ⊗ Fp k(( )))-Mod dd of étale (φ, F ⊗ Fp k(( )))-modules with descent data: an object M is defined as for the category (φ, F ⊗ Fp k((p)))-Mod but we moreover require that M is endowed with a semilinear action of Gal(K ∞ /(K 0 ) ∞ ) (semilinear with respect to the residual action on F ⊗ Fp k(( )) where F is endowed with the trivial Gal(K ∞ /(K 0 ) ∞ )-action) commuting with φ.

By work of Fontaine [START_REF] Fontaine | Représentations p-adiques des corps locaux. II[END_REF], there are anti-equivalences

φ, F ⊗ Fp k((p)) -Mod ∼ -→ Rep F (G (K0)∞ ) and φ, F ⊗ Fp k(( )) -Mod dd ∼ -→ Rep F (G (K0)∞ )
given by

M -→ Hom M, k((p)) sep and M → Hom (M, k(( )) sep )
respectively. See also [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], Appendix A.2.

The following proposition summarizes the relation between the various categories and functors we introduced above. Proposition 3.2.24 ([HLM17], Proposition 2.2.9). There exist faithful functors

M k(( )) : F-BrMod r dd → φ, F ⊗ Fp k(( )) -Mod dd and F : F-FLMod [0,p-2] → φ, F ⊗ Fp k((p)) -Mod
fitting in the following commutative diagram:

F-BrMod r dd T * st M k(( )) / / φ, F ⊗ Fp k(( )) -Mod dd Hom(-,k(( )) sep ) y y Rep F (G K0 ) Res / / Rep F (G (K0)∞ ) F-FLMod [0,p-2] T * cris O O F / / φ, F ⊗ Fp k((p)) -Mod -⊗ k((p)) k(( )) O O Hom(-,k((p)) sep ) e e
where the descent data is relative to K 0 and the functor Res • T * cris is fully faithful.

Note that the functors M k(( )) and F are defined in [START_REF] Breuil | Formes modulaires de Hilbert modulo p et valeurs d'extension entre caractères galoisiens[END_REF]. (See also [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], Appendix A). The following is an immediate consequence of Proposition 3.2.24, which is also stated in [LMP], Corollary 2.14.

Corollary 3.2.25. Let 0 ≤ r ≤ p -2, and let M (resp. M ) be an object in F-BrMod r dd (resp. in

F-FLMod [0,p-2] ). Assume that T * st (M) is Fontaine-Laffaille. If M k(( )) (M) ∼ = F(M ) ⊗ k((p)) k(( ))
then one has an isomorphism of G K0 -representations

T * st (M) ∼ = T * cris (M ).
The following two lemmas are very crucial in this paper, as we will see later, which describe the functors M k(( )) and F respectively. Lemma 3.2.26 ([HLM17], Lemma 2.2.6). Let M be a Breuil module of inertial type n-1 i=0 ω ki with a framed basis e for M and a framed system of generators f for Fil r M, and write M * for its dual as defined in Definition 3.2.7. Let V = Mat e,f (Fil r M) ∈ M , n (S) and A = Mat e,f (φ r ) ∈ GL n (S). Then there exists a basis e for M k((

)) (M * ) with g •e i = (ω -p -1 ki (g)⊗1)e i for all i ∈ {0, 1, • • • , n- 1} and g ∈ Gal(K/K 0 ), such that the Frobenius φ on M k(( )) (M * ) is described by Mat e (φ) = V t A -1 t ∈ M n (F ⊗ Fp k[[ ]]) where V , A are lifts of V, A in M n (F ⊗ Fp k[[ ]]) via the reduction morphism F ⊗ Fp k[[ ]] S induced by → u and Mat e (φ) i,j ∈ F ⊗ Fp k[[ ]] ω p -1 k i -k j .
Lemma 3.2.27 ([HLM17], Lemma 2.2.7). Let M ∈ F-FLMod [0,p-2] be a rank n Fontaine-Laffaille module with parallel Hodge-Tate weights 0 ≤ m 0 ≤ • • • ≤ m n-1 ≤ p -2 (counted with multiplicity). Let e = (e 0 , . . . , e n-1 ) be a k ⊗ Fp F basis for M , compatible with the Hodge filtration Fil • M and let F ∈ M n (k ⊗ Fp F) be the associated matrix of the Frobenius φ

• : gr • M → M .
Then there exists a basis e for M := F(M ) such that the Frobenius φ on M is described by

Mat e (φ) = Diag p m0 , • • • , p mn-1 • F ∈ M n (F ⊗ Fp k[[p]]).

Local Galois side

In this section, we study ordinary Galois representations and their potentially crystalline lifts. In particular, we prove that the Frobenius eigenvalues of certain potentially crystalline lifts preserve the information of the wildly ramified part of ordinary representations. Throughout this section, we let f be a positive integer, K = Q p , e = p f -1, and

K = Q p f ( e √ -p
). We also fix := e √ -p, and let S = (F

p f ⊗ Fp F)[u]/u ep and S 0 := S ω 0 f = (F p f ⊗ Fp F)[u e ]/u ep ⊆ S. Recall that by [m] f for a rational number m ∈ Z[ 1 p ]
we mean the unique integer in [0, e) congruent to m mod (e).

We say that a representation ρ 0 : G Qp → GL n (F) is ordinary if it is isomorphic to a representation whose image is contained in the Borel subgroup of upper-triangular matrices. Namely, an ordinary representation has a basis e := (e n-1 , e n-2 , • • • , e 0 ) that gives rise to a matrix form as follows:

ρ 0 ∼ =        U µn-1 ω cn-1+(n-1) * n-1 • • • * * 0 U µn-2 ω cn-2+(n-2) • • • * * . . . . . . . . . . . . . . . 0 0 • • • U µ1 ω c1+1 * 1 0 0 • • • 0 U µ0 ω c0        (3.3.1)
Here, U µ is the unramified character sending the geometric Frobenius to µ ∈ F × and c i are integers. By ρ 0 , we always mean an n-dimensional ordinary representation that is written as in (3.3.1). For n -1 ≥ i ≥ j ≥ 0, we write

ρ i,j (3.3.2)
for the (i -j + 1)-dimensional subquotient of ρ 0 determined by the subset (e i , e i-1 , • • • , e j ) of the basis e. For instance, ρ i,i = U µi ω ci+i and ρ n-1,0 = ρ 0 . An ordinary representation G Qp → GL n (F) is maximally non-split if its socle filtration has length n. For instance, ρ 0 in (3.3.1) is maximally non-split if and only if

* i = 0 for all i = 1, 2, • • • , n -1.
In this paper, we are interested in ordinary maximally non-split representations satisfying a certain genericity condition.

Definition 3.3.3. We say that ρ 0 is generic if c i+1 -c i > n -1 for all i ∈ {0, 1, • • • , n -2} and c n-1 -c 0 < (p -1) -(n -1).
We say that ρ 0 is strongly generic if ρ 0 is generic and

c n-1 -c 0 < (p -1) -(3n -5).
Note that this strongly generic condition implies p > n 2 + 2(n -3). We describe a rough shape of the Breuil modules with descent data from K to K = Q p corresponding to ρ 0 . Let r be a positive integer with p-1 > r ≥ n-1, and let M ∈ F-BrMod r dd be a Breuil module of inertial type n-1 i=0 ω ki f such that T r st (M) ∼ = ρ 0 , for some k i ∈ Z. By Proposition 3.2.9, we note that M is a successive extension of M i , where M i := M(k i , r i , ν i ) (c.f. Lemma 3.2.10) is a rank one Breuil module of inertial type ω ki f such that

ω ki+pri f ∼ = T r st (M i )| I Qp ∼ = ω ci+i (3.3.4) for each i ∈ {0, 1, • • • , n -1}.
More precisely, there exist a framed basis e = (e n-1 , e n-2 , • • • , e 0 ) for M and a framed system of generators

f = (f n-1 , f n-2 , • • • , f 0 ) for Fil r M such that Mat e,f (Fil r M) =      u rn-1(p-1) u [p -1 kn-2-kn-1] f v n-1,n-2 • • • u [p -1 k0-kn-1] f v n-1,0 0 u rn-2(p-1) • • • u [p -1 k0-kn-2] f v n-2,0 . . . . . . . . . . . . 0 0 • • • u r0(p-1)      , (3.3.5) Mat e,f (φ r ) =      ν n-1 u [kn-2-kn-1] f w n-1,n-2 • • • u [k0-kn-1] f w n-1,0 0 ν n-2 • • • u [k0-kn-2] f w n-2,0 . . . . . . . . . . . . 0 0 • • • ν 0      , (3.3.6)
and

Mat e (N ) =        0 u [kn-2-kn-1] f γ n-1,n-2 • • • u [k1-kn-1] f γ n-1,1 u [k0-kn-1] f γ n-1,0 0 0 • • • u [k1-kn-2] f γ n-2,1 u [k0-kn-2] f γ n-2,0 . . . . . . . . . . . . . . . 0 0 • • • 0 u [k0-k1] f γ 1,0 0 0 • • • 0 0        (3.3.7)
for some ν i ∈ (F p f ⊗ Fp F) × and for some v i,j , w i,j , γ i,j ∈ S 0 . Fix 0 ≤ j ≤ i ≤ n -1. We define the Breuil submodule

M i,j (3.3.8)
that is a subquotient of M determined by the basis (e i , e i-1 , • • • , e j ). For instance, M i,i ∼ = M i for all 0 ≤ i ≤ n -1. We note that T r st (M i,j ) ∼ = ρ i,j by Proposition 3.2.9. We will keep these notation and assumptions for M throughout this paper.

Elimination of Galois types

In this section, we find out the possible Galois types of niveau 1 for potentially semi-stable lifts of ρ 0 with Hodge-Tate weights

{-(n -1), -(n -2), • • • , 0}.
We start this section with the following elementary lemma.

Lemma 3.3.9. Let ρ : G Qp → GL n (E) be a potentially semi-stable representation with Hodge-Tate weights {-(n -1), ..., -2, -1, 0} and of Galois type

n-1 i=0 ω ki f . Then det(ρ)| I Qp = ε n(n-1) 2 • ω n-1 i=0 ki f
, where ε is the cyclotomic character.

Proof. det(ρ) is a potentially crystalline character of G Qp with Hodge-Tate weight -(

n-1 i=0 i) and of Galois type ω n-1 i=0 ki f , i.e., det(ρ) • ω -n-1 i=0 ki f is a crystalline character with Hodge-Tate weight -( n-1 i=0 i) = -n(n-1) 2 so that det(ρ)| I Qp • ω -n-1 i=0 ki f ∼ = ε n(n-1) 2 .
We will only consider the Breuil modules M corresponding to the mod p reduction of the strongly divisible modules that corresponds to the Galois stable lattices in potentially semi-stable lifts of ρ 0 with Hodge-Tate weights {-(n -1), -(n -2), • • • , -1, 0}, so that we may assume that r = n -1, i.e., M ∈ F-BrMod n-1 dd .

Lemma 3.3.10. Let f = 1. Assume that ρ 0 is generic, and that M ∈ F-BrMod n-1 dd corresponds to the mod p reduction of a strongly divisible module M such that T n-1 st (M) ∼ = ρ 0 and T Qp,n-1 st ( M) is a Galois stable lattice in a potentially semi-stable lift of ρ 0 with Hodge-Tate weights {-(n -1), -(n -2), • • • , 0} and Galois type n-1 i=0 ω ki for some integers k i . Then there exists a framed basis e for M and a framed system of generators f for Fil n-1 M such that Mat e,f (Fil n-1 M), Mat e,f (φ n-1 ), and Mat e (N ) are as in (3.3.5), (3.3.6), and (3.3.7) respectively. Moreover, the (k i , r i ) satisfy the following properties:

(i) k i ≡ c i + i -r i mod (e) for all i ∈ {0, 1, • • • , n -1}; (ii) 0 ≤ r i ≤ n -1 for all i ∈ {0, 1, • • • , n -1}; (iii) n-1 i=0 r i = (n-1)n 2 .
Proof. Note that the inertial type of M is n-1 i=0 ω ki by Proposition 3.2.18. The first part of the Lemma is obvious from the discussion at the beginning of Section 3.3.

We now prove the second part of the Lemma. We may assume that the rank-one Breuil modules M i are of weight n -1, so that 0 ≤ r i ≤ n -1 for i = {0, 1, ..., n -1} by Lemma 3.2.10. By the equation (3.3.4), we have k i ≡ c i + i -r i mod (e), as e = p -1. By looking at the determinant of ρ 0 we deduce the conditions

ω n(n-1) 2 +kn-1+kn-2+•••+k0 = det T n-1 st (M)| I Qp = det ρ 0 | I Qp = ω cn-1+cn-2+•••+c0+ n(n-1)
2 from Lemma 3.3.9, and hence we have

r n-1 + r n-2 + • • • + r 0 = n(n-1)

2

(as p > n 2 + 2(n -3) due to the genericity of ρ 0 ).

One can further eliminate Galois types of niveau 1 if ρ 0 is maximally non-split.

Proposition 3.3.11. Keep the assumptions and notation of Lemma 3.3.10. If the tuple (k i , r i ) further satisfy one of the following conditions

• r i = n -1 for some i ∈ {0, 1, 2, • • • , n -2}; • r i = 0 for some i ∈ {1, 2, 3, • • • , n -1}, then ρ 0 is not maximally non-split.
Proof. The main ingredient is Lemma 3.2.11. We fix i ∈ {0, 1, 2, • • • , n -2} and identify x = i + 1 and y = i and all the other following. From the results in Lemma 3.3.10, it is easy to compute that

[k i -k i+1 ] 1 = e-(c i+1 -c i +1)+(r i+1 -r i )
. By the genericity conditions in Definition 3.3.3 and by part (ii) of Lemma 3.3.10, we see that 0 < [k i -k i+1 ] 1 < e so that if r i ≥ r i+1 then the equation (3.2.12) in Lemma 3.2.11 holds.

If

r i+1 e ≤ [k i -k i+1 ] 1 and r i ≥ r i+1 , then * i+1 = 0 by Lemma 3.2.11. Since 0 < [k i -k i+1 ] 1 < e, we have r i+1 e ≤ [k i -k i+1 ] 1 if
and only if r i+1 = 0, in which case ρ 0 is not maximally non-split.

We now apply the second part of Lemma 3.2.11. It is easy to check that j 0 = r i+1 -1. One can again readily check that the equation (3.2.15) is equivalent to r i = n -1, in which case * i+1 = 0 so that ρ 0 is not maximally non-split.

Note that all of the Galois types that will appear later in this section will satisfy the conditions in Lemma 3.3.10, and Proposition 3.3.11 as well if we further assume that ρ 0 is maximally non-split.

Fontaine-Laffaille parameters

In this section, we parameterize the wildly ramified part of generic and maximally non-split ordinary representations using Fontaine-Laffaille theory.

We start this section by recalling that if ρ 0 is generic then ρ 0 ⊗ ω -c0 is Fontaine-Laffaille (c.f. [GG10], Lemma 3.1.5), so that there is a Fontaine-Laffaille module M with Hodge-Tate weights

{0, c 1 -c 0 + 1, • • • , c n-1 -c 0 + (n -1)} such that T * cris (M ) ∼ = ρ 0 ⊗ ω -c0
(if we assume that ρ 0 is generic).

Lemma 3.3.12. Assume that ρ 0 is generic, and let M ∈ F-FLMod [0,p-2] Fp be a Fontaine-Laffaille module such that T * cris (M ) ∼ = ρ 0 ⊗ ω -c0 . Then there exists a basis e = (e 0 , e 1 , • • • , e n-1 ) for M such that

Fil j M =    M if j ≤ 0; F(e i , • • • , e n-1 ) if c i-1 -c 0 + i -1 < j ≤ c i -c 0 + i; 0 if c n-1 -c 0 + n -1 < j.
and

Mat e (φ • ) =          µ -1 0 α 0,1 α 0,2 • • • α 0,n-2 α 0,n-1 0 µ -1 1 α 1,2 • • • α 1,n-2 α 1,n-1 0 0 µ -1 2 • • • α 2,n-2 α 2,n-1 . . . . . . . . . . . . . . . . . . 0 0 0 • • • µ -1 n-2 α n-2,n-1 0 0 0 • • • 0 µ -1 n-1          (3.3.13)
where α i,j ∈ F.

Note that the basis e on M in Lemma 3.3.12 is compatible with the filtration.

Proof. This is an immediate generalization of [HLM17], Lemma 2.1.7.

For i ≥ j, the subset (e j , • • • , e i ) of e determines a subquotient M i,j of the Fontaine-Laffaille module M , which is also a Fontaine-Laffaille module with the filtration induced from Fil s M in the obvious way and with Frobenius described as follows:

A i,j :=        µ -1 j α j,j+1 • • • α j,i-1 α j,i 0 µ -1 j+1 • • • α j+1,i-1 α j+1,i . . . . . . . . . . . . . . . 0 0 • • • µ -1 i-1 α i-1,i 0 0 • • • 0 µ -1 i        . Note that T * cris (M i,j ) ⊗ ω c0 ∼ = ρ i,j
. We let A i,j be the (i -j) × (i -j)-submatrix of A i,j obtained by deleting the left-most column and the lowest row of A i,j . Lemma 3.3.14. Keep the assumptions and notation of Lemma 3.3.12, and let 0 ≤ j < j + 1 < i ≤ n -1. Assume further that ρ 0 is maximally non-split.

If

det A i,j = (-1) i-j+1 µ -1 j+1 • • • µ -1 i-1 α j,i
, then [α j,i : det A i,j ] ∈ P 1 (F) does not depend on the choice of basis e compatible with the filtration.

Proof. This is an immediate generalization of [HLM17], Lemma 2.1.9. Definition 3.3.15. Keep the assumptions and notation of Lemma 3.3.14, and assume further that

ρ 0 satisfies det A i,j = (-1) i-j+1 µ -1 j+1 • • • µ -1 i-1 α j,i (3.3.16)
for all i, j ∈ Z with 0 ≤ j < j + 1 < i ≤ n -1. The Fontaine-Laffaille parameter associated to ρ 0 is defined as

FL n (ρ 0 ) := FL i,j n (ρ 0 ) i,j ∈ [P 1 (F)] (n-2)(n-1) 2
where FL i,j n (ρ 0 ) := α j,i : (-1) i-j+1 • det A i,j ∈ P 1 (F) for all i, j ∈ Z such that 0 ≤ j < j + 1 < i ≤ n -1.

We often write y

x for [x : y] ∈ P 1 (F) if x = 0. The conditions in (3.3.16) for i, j guarantee the well-definedness of FL i,j n (ρ 0 ) in P 1 (F). We also point out that FL i,j n (ρ

0 ) = (-1) i-j µ -1 j+1 • • • µ -1 i-1 in P 1 (F).
One can define the inverses of the elements in P 1 (F) in a natural way: for [x 1 :

x 2 ] ∈ P 1 (F), [x 1 : x 2 ] -1 := [x 2 : x 1 ] ∈ P 1 (F).
Lemma 3.3.17. Assume that ρ 0 is generic. Then

(i) ρ ∨ 0 is generic;
(ii) if ρ 0 is strongly generic, then so is ρ ∨ 0 ;

(iii) if ρ 0 is maximally non-split, then so is ρ ∨ 0 ;

(iv) if ρ 0 is maximally non-split, then the conditions in (3.3.16) are stable under ρ 0 → ρ ∨ 0 . Assume further that ρ 0 is maximally non-split and satisfies the conditions in (3.3.16).

(v) for all i, j ∈ Z with 0 ≤ j < j + 1 < i ≤ n -1, FL i,j n (ρ 0 ) = FL i,j n (ρ 0 ⊗ ω b ) for any b ∈ Z; (vi) for all i, j ∈ Z with 0 ≤ j < j + 1 < i ≤ n -1, FL i,j n (ρ 0 ) = FL i-j,0 i-j+1 (ρ i,j );
(vii) for all i, j ∈ Z with 0 ≤ j < j + 1 < i ≤ n -1, FL i,j n (ρ 0 ) -1 = FL n-1-j,n-1-i n (ρ ∨ 0 ). Proof. (i), (ii) and (iii) are easy to check. We leave them for the reader.

The only effect on Fontaine-Laffaille module by twisting ω b is shifting the jumps of the filtration. Thus (v) and (vi) are obvious.

For (iv) and (vii), one can check that the Frobenius of the Fontaine-Laffaille module associated to ρ ∨ 0 is described by

       0 0 • • • 0 1 0 0 • • • 1 0 . . . . . . . . . . . . . . . 0 1 • • • 0 0 1 0 • • • 0 0        • [Mat e (φ • ) t ] -1 •        0 0 • • • 0 1 0 0 • • • 1 0 . . . . . . . . . . . . . . . 0 1 • • • 0 0 1 0 • • • 0 0       
where Mat e (φ • ) is as in (3.3.13). Now one can check them by direct computation.

We end this section by defining certain numerical conditions on Fontaine-Laffaille parameters. We consider the matrix (1, n)w 0 Mat e (φ • ) t , where Mat e (φ • ) is the upper-triangular matrix in (3.3.13).

Here, w 0 is the longest element of the Weyl group W associated to T and (1, n) is a permutation in W . Note that the anti-diagonal matrix displayed in the proof of Lemma 3.3.17 is w 0 seen as an element in GL n (F). For 1 ≤ i ≤ n -1 we let B i be the square matrix of size i that is the left-bottom corner of (1, n)w 0 Mat e (φ • ) t . Definition 3.3.18. Keep the notation and assumptions of Definition 3.3.15. We say that ρ 0 is Fontaine-Laffaille generic if moreover det B i = 0 for all 1 ≤ i ≤ n -1 and ρ 0 is strongly generic.

We emphasize that by an ordinary representation ρ 0 being Fontaine-Laffaille generic, we always mean that ρ 0 satisfies the maximally non-splitness and the conditions in (3.3.16) as well as det B i = 0 for all 1 ≤ i ≤ n -1 and the strongly generic assumption (c.f. Definition 3.3.3).

Although the Frobenius matrix of a Fontaine-Laffaille module depends on the choice of basis, it is easy to see that the non-vanishing of the determinants above is independent of the choice of basis compatible with the filtration. Note that the conditions in Definition 3.3.18 are necessary and sufficient conditions for

(1, n)w 0 Mat e (φ • ) t ∈ B(F)w 0 B(F)
in the Bruhat decomposition, which will significantly reduce the size of the paper (c.f. Remark 3.3.19).

We also note that

• det B 1 = 0 if and only if FL n-1,0 n (ρ 0 ) = ∞; • det B n-1 = 0 if and only if FL n-1,0 n (ρ 0 ) = 0.
Finally, we point out that the locus of Fontaine-Laffaille generic ordinary Galois representations ρ 0 forms a (Zariski) open subset in [P 1 (F)]

(n-1)(n-2) 2

.

Remark 3.3.19. Definition 3.3.18 comes from the fact that the list of Serre weights of ρ 0 is then minimal in the sense of Conjecture 3.5.16. It is very crucial in the proof of Theorem 3.5.44 as it is more difficult to track the Fontaine-Laffaille parameters on the automorphic side if we have too many Serre weights. Moreover, these conditions simplify our proof for Theorem 3.3.44.

Breuil modules of certain inertial types of niveau 1

In this section, we classify the Breuil modules with certain inertial types, corresponding to the ordinary Galois representations ρ 0 as in (3.3.1), and we also study their corresponding Fontaine-Laffaille parameters.

Throughout this section, we always assume that ρ 0 is strongly generic. Since we are only interested in inertial types of niveau 1, we let f = 1, e = p -1, and = e √ -p. We define the following integers for 0 ≤ i ≤ n -1:

r (0) i :=    1 if i = n -1; i if 0 < i < n -1; n -2 if i = 0. (3.3.20)
We also set k

(0) i := c i + i -r (0) i for all i ∈ {0, 1, • • • , n -1}.
We first classify the Breuil modules of inertial types described as above.

Lemma 3.3.21. Assume that ρ 0 is strongly generic and that M ∈ F-BrMod n-1 dd corresponds to the mod p reduction of a strongly divisible modules M such that T Qp,n-1 st ( M) is a Galois stable lattice in a potentially semi-stable lift of ρ 0 with Hodge-Tate weights

{-(n -1), -(n -2), • • • , 0} and Galois type n-1 i=0 ω k (0)
i . Then M ∈ F-BrMod n-1 dd can be described as follows: there exist a framed basis e for M and a framed system of generators f for Fil n-1 M such that

Mat e,f (Fil n-1 M) =       u r (0) n-1 e β n-1,n-2 u r (0) n-1 e-k (0) n-1,n-2 • • • β n-1,0 u r (0) n-1 e-k (0) n-1,0 0 u r (0) n-2 e • • • β n-2,0 u r (0) n-2 e-k (0) n-2,0 . . . . . . . . . . . . 0 0 • • • u r (0) 0 e       and Mat e,f (φ n-1 ) = Diag (ν n-1 , ν n-2 , • • • , ν 0 )
where k

(0) i,j := k (0) i -k (0) j , ν i ∈ F × and β i,j ∈ F. Moreover, Mat e (N ) = γ i,j • u [k (0) j -k (0) i ]1
where γ i,j = 0 if i ≤ j and γ i,j ∈ u e[k (0)

j -k (0) i ]1 S 0 if i > j.
Note that e and f in Lemma 3.3.21 are not necessarily the same as the ones in Lemma 3.3.10.

Proof. We keep the notation in (3.

if n -1 ≥ i > j ≥ 0 then e -(k (0) i -k (0) j ) ≥ n. (3.3.22)
Indeed, by the strongly generic assumption, Definition 3.3.3

e -(k

(0) i -k (0) j ) = (p -1) -(c i + i -r (0) i ) + (c j + j -r (0) j ) = (p -1) -(c i -c j ) -(i -j) + (r (0) i -r (0) j ) ≥ (p -1) -(c n-1 -c 0 ) -(n -1 -0) + (1 -(n -2)) ≥ 3n -4 -2n + 4 = n.
Note that this claim will be often used during the proof later.

We now diagonalize Mat e,f (φ n-1 ) with some restriction on the powers of the entries of the matrix Mat e,f (Fil n-1 M). Let V 0 = Mat e,f (Fil n-1 M) ∈ M n (S) and A 0 = Mat e,f (φ n-1 ) ∈ GL n (S). We also let V 1 ∈ M n (S) be the matrix obtained from V 0 by replacing v i,j by v i,j ∈ S 0 , and B 1 ∈ GL n (S) the matrix obtained from A 0 by replacing w i,j by w i,j ∈ S 0 . It is straightforward to check that

A 0 • V 1 = V 0 • B 1 if and only if for all i > j ν i v i,j u [k (0) j -k (0) i ]1 + i-1 s=j+1 w is v s,j u [k (0) s -k (0) i ]1+[k (0) j -k (0) s ]1 + w i,j u r (0) j e+[k (0) j -k (0) i ]1 = w i,j u r (0) i e+[k (0) j -k (0) i ]1 + i-1 s=j+1 v i,s w s,j u [k (0) s -k (0) i ]1+[k (0) j -k (0) s ]1 + ν j v i,j u [k (0) j -k (0) i ]1 . (3.3.23) Note that the power of u in each term of (3.3.23) is congruent to [k (0) j -k (0)
i ] 1 modulo (e). It is immediate that for all i > j there exist v i,j ∈ S 0 and w i,j ∈ S 0 satisfying the equation (3.3.23) with the following additional properties: for all i > j

deg v i,j < r (0) i e.
(3.3.24)

Letting e := eA 0 , we have

Mat e ,f (Fil n-1 M) = V 1 and Mat e ,f (φ n-1 ) = φ(B 1 )
where f = e V 1 , by Lemma 3.2.19. Note that φ(B 1 ) is congruent to a diagonal matrix modulo (u ne ) by (3.3.22). We repeat this process one more time. We may assume that w i,j ∈ u ne S 0 , i.e., that

A 0 ≡ B 1 modulo (u ne )
where B 1 is assumed to be a diagonal matrix. It is obvious that there exists an

upper-triangular matrix V 1 = (v i,j u [p -1 k (0) j -k (0)
i ]2 ) whose entries have bounded degrees as in (3.3.24), satisfying the equation A 0 V 1 ≡ V 0 B 1 modulo (u ne ). By Lemma 3.2.19, we get Mat e ,f (φ n-1 ) is diagonal. Hence, we may assume that Mat e,f (φ n-1 ) is diagonal and that deg v i,j in Mat e,f (Fil n-1 M) is bounded as in (3.3.24), and we do so. Moreover, this change of basis do not change the shape of Mat e (N ), so that we also assume that Mat e (N ) is still as in (3.3.7).

We now prove that

v i,j u [k (0) j -k (0) i ]1 = β i,j u r (0) i e-(k (0)
i -k

(0) j ) (3.3.25)
for all n -1 ≥ i > j ≥ 0, where β i,j ∈ F. Note that this is immediate for i = n -1 and i = 1, since r

(0) i = 1 if i = n -1 or i = 1.
To prove (3.3.25), we induct on i. The case i = 1 is done as above. Fix p 0 ∈ {2, 3, • • • , n -2}, and assume that (3.3.25) holds for all i ∈ {1, 2, • • • , p 0 -1} and for all j < i. We consider the subquotient M p0,0 of M defined in (3.3.8). By abuse of notation, we write e = (e p0 , • • • , e 0 ) for the induced framed basis for M p0,0 and f = (f p0 , • • • , f 0 ) for the induced framed system of generators for Fil n-1 M p0,0 .

We claim that for p 0 ≥ j ≥ 0

u e N (f j ) ∈ S 0 u e f j + p0 t=j+1 S 0 u [k (0) j -k (0) t ]1 f t .
Consider N (f j ) = N (f j -u r (0) j e e j ) + N (u r (0) j e e j ). It is easy to check that N (f j -u r (0) j e e j ) and N (u r (0) j e e j ) + r (0) j ef j are S-linear combinations of e n-1 , • • • , e j+1 , and they are, in fact, S 0 -linear combinations of u

[k (0) j -k (0) n-1 ] e n-1 , • • • , u [k (0) j -k (0) j+1 ] e j+1 since they are ω k (0) j -invariant. Since u e N (f j ) ∈ Fil n-1 M ⊃ u (n-1)e M and u e N (f j ) + r (0) j eu e f j = [N (f j -u r (0)
j e e j )] + [N (u r (0) j e e j ) + r (0) j ef j ], we conclude that

u e N (f j ) + r (0) j eu e f j ∈ p0 t=j+1 S 0 u [k (0) j -k (0) t ]1 f t , which completes the claim. Let Mat e,f (N | Mp 0 ,0 ) = γ i,j • u [k (0) j -k (0) i ]1 where γ i,j = 0 if i ≤ j and γ i,j ∈ S 0 if i > j. We also claim that γ i,j ∈ u e[k (0) j -k (0)
i ]1 S 0 for p 0 ≥ i > j ≥ 0, which can be readily checked from the equation cN φ n-1 (f j ) = φ n-1 (u e N (f j )).

(Note that c = 1 ∈ S as E(u) = u e + p.) Indeed, we have

cN φ n-1 (f j ) = N (ν j e j ) = ν j p0 i=j+1 γ i,j u [k (0) j -k (0) i ]1 e i .
On the other hand, since Mat e,f (φ n-1 | Mp 0 ,0 ) is diagonal, the previous claim immediately implies that

φ r (u e N (f j )) ∈ p0 t=j+1 S 0 u p[k (0) j -k (0) t ]1 e t .
Hence, we conclude the claim.

We now finish the proof of (3.3.25) by inducting on p 0 -j as well. Write

v i,j = r (0) i -1 t=0 x (t) i,j u te for x (t) i,j ∈ F. We need to prove x (t) p0,j = 0 for t ∈ {0, 1, • • • , r (0)
p0 -2}. Assume first j = p 0 -1, and we compute N (f j ) as follows:

N (f p0-1 ) = - r (0) p 0 -1 t=0 x (t) p0,p0-1 [e(t + 1) -(k (0) p0 -k (0) p0-1 )]u e(t+1)-(k (0) p 0 -k (0) p 0 -1 ) e p0 + γ p0,p0-1 u (r (0) p 0 -1 +1)e-(k (0) p 0 -k (0) p 0 -1 ) e p0 -r (0)
p0-1 eu r (0) p 0 -1 e e p0-1 , which immediately implies

N (f p0-1 ) ≡ r (0) p 0 -1 t=0 x (t) p0,p0-1 [er (0) p0-1 -e(t + 1) + (k (0) p0 -k (0) p0-1 )]u e(t+1)-(k (0) p 0 -k (0) p 0 -1 ) e p0 + γ p0,p0-1 u (r (0) p 0 -1 +1)e-(k (0) p 0 -k (0) p 0 -1 ) e p0 modulo Fil n-1 M p0,0 . Since γ p0,p0-1 ∈ u e[e-(k (0) p 0 -k (0)
p 0 -1 )] S 0 and e -(k

(0) p0 -k (0)
p0-1 ) ≥ n by (3.3.22), we get

N (f p0-1 ) ≡ r (0) p 0 -1 t=0 x (t) p0,p0-1 [er (0) p0-1 -e(t + 1) + (k (0) p0 -k (0) p0-1 )]u e(t+1)-(k (0) p 0 -k (0)
p 0 -1 ) e p0 modulo Fil n-1 M p0,0 , so that

u e N (f p0-1 ) ≡ r (0) p 0 -2 t=0 x (t) p0,p0-1 [er (0) p0-1 -e(t + 1) + (k (0) p0 -k (0) p0-1 )]u e(t+2)-(k (0) p 0 -k (0)
p 0 -1 ) e p0 modulo Fil n-1 M p0,0 .

It is easy to check that er

(0) p0-1 -e(t + 1) + (k (0) p0 -k (0) p0-1 ) ≡ 0 (3.3.26) modulo (p) for all 0 ≤ t ≤ r (0) p0 -2. Indeed, er (0)
p0-1 -e(t + 1) + (k

(0) p0 -k (0) p0-1 ) ≡ -r (0) p0-1 + (t + 1) + (k (0) p0 -k (0) p0-1 ) = (t + 1) + (c p0 -c p0-1 + 1) -r (0) p0 modulo (p). Since 0 ≤ t ≤ r (0) p0 -2, 0 < (c p0 -c p0-1 + 2) -r (0) p0 ≤ (t + 1) + (c p0 -c p0-1 + 1) -r (0) p0 ≤ (c p0 -c p0-1 -1) < p
by the strongly generic conditions, Definition 3.3.3. Hence, we conclude that x

(t) p0,p0-1 = 0 for all 0 ≤ t ≤ r (0) p0 -2 since u e N (f p0-1 ) ∈ Fil n-1 M p0,0
. This completes the proof of (3.3.25) for j = p 0 -1. Assume that (3.3.25) holds for i = p 0 and j ∈ {p 0 -1, p 0 -2, • • • , s + 1}. We compute N (f s ) for p 0 -1 > s ≥ 0 as follows: using the induction hypothesis on i ∈ {1, 2,

• • • , p 0 -1} N (f s ) = - r (0) p 0 -1 t=0 x (t) p0,s [e(t + 1) -(k (0) p0 -k (0) s )]u e(t+1)-(k (0) p 0 -k (0) s ) e p0 + p0-1 i=s+1 β i,s u r (0) i e-(k (0) i -k (0) s ) p0 s=i+1 γ s,i u e-(k (0) s -k (0) i ) e s -[r (0) i e -(k (0) i -k (0) s )]e i + u r (0) s e p0 i=s+1 γ i,s u e-(k (0) i -k (0)
s ) e i -r (0) s eu r (0) s e e s .

Since

γ i,j ∈ u e[e-(k (0) i -k (0) j )] S 0 , we have N (f s ) ≡ - r (0) p 0 -1 t=0 x (t) p0,s [e(t + 1) -(k (0) p0 -k (0) s )]u e(t+1)-(k (0) p 0 -k (0) s ) e p0 - p0-1 i=s+1 β i,s [r (0) i e -(k (0) i -k (0) s )]u r (0) i e-(k (0) i -k (0)
s ) e i -r (0) s eu r (0) s e e s modulo Fil n-1 M p0,0 , which immediately implies

N (f s ) ≡ r (0) p 0 -1 t=0 x (t) p0,s [r (0) s e -e(t + 1) + (k (0) p0 -k (0) s )]u e(t+1)-(k (0) p 0 -k (0) s ) e p0 + p0-1 i=s+1 β i,s [r (0) s e -r (0)
i e + (k

(0) i -k (0) s )]u r (0) i e-(k (0) i -k (0)
s ) e i modulo Fil n-1 M p0,0 . Now, from the induction hypothesis on j

∈ {p 0 -1, p 0 -2, • • • , s + 1}, u e p0-1 i=s+1 β i,s [r (0) s e -r (0)
i e + (k

(0) i -k (0) s )]u r (0) i e-(k (0) i -k (0) s ) e i ∈ Fil n-1 M p0,0
and so we have

u e N (f s ) ≡ r (0) p 0 -2 t=0 x (t) p0,s [r (0) s e -e(t + 1) + (k (0) p0 -k (0) s )]u e(t+2)-(k (0) p 0 -k (0)
s ) e p0 modulo Fil n-1 M p0,0 . By the same argument as (3.3.26), one can readily check that r Then

β i,i-1 ∈ F × for i ∈ {1, 2, • • • , n -1}
and we have the following identities: for 0

≤ j < j + 1 < i ≤ n -1 FL i,j n (ρ 0 ) = β i,j ν j+1 • • • ν i-1 : (-1) i-j+1 det A i,j ∈ P 1 (F) where A i,j =          β j+1,j β j+2,j β j+3,j • • • β i-1,j β i,j 1 β j+2,j+1 β j+3,j+1 • • • β i-1,j+1 β i,j+1 0 1 β j+3,j+2 • • • β i-1,j+2 β i,j+2 . . . . . . . . . . . . . . . . . . 0 0 0 • • • β i-1,i-2 β i,i-2 0 0 0 • • • 1 β i,i-1          .
Proof. We may assume c 0 = 0 by Lemma 3.3.17. We let V := Mat e,f (Fil n-1 M) and A := Mat e,f (φ n-1 ) be as in the statement of Lemma 3.3.21. By Lemma 3.2.26, the φ-module over F ⊗ Fp F p (( )) defined by M := M Fp(( )) (M * ) is described as follows:

Mat e (φ) = (U i,j )

where

U i,j =      ν -1 j • r (0) j e if i = j; 0 if i > j; ν -1 j • β j,i • r (0) j e-(k (0) j -k (0) i ) if i < j in a framed basis e = (e n-1 , e n-2 , • • • , e 0 ) with dual type ω -k (0) n-1 ⊕ ω -k (0) n-2 • • • ⊕ ω -k (0) 0 . By considering the change of basis e = ( k (0) n-1 e n-1 , k (0) n-2 e n-2 , • • • , k (0) 0 e 0 )
, Mat e (φ) is described as follows:

Mat e (φ) = (V i,j )

where

V i,j =      ν -1 j • e(k (0) j +r (0) j ) if i = j; 0 if i > j; ν -1 j • β j,i • e(k (0) j +r (0) j ) if i < j. Since k (0) i = c i + i -r (0) i for each n -1 ≥ i ≥ 0, we easily see that the φ-module M 0 is the base change via F ⊗ Fp F p ((p)) → F ⊗ Fp F p (( )) of the φ-module M 0 over F ⊗ Fp F p ((p)) described by Mat e (φ) =      ν -1 n-1 p cn-1+(n-1) 0 • • • 0 ν -1 n-1 β n-1,n-2 p cn-1+(n-1) ν -1 n-2 p cn-2+(n-2) • • • 0 . . . . . . . . . . . . ν -1 n-1 β n-1,0 p cn-1+(n-1) ν -1 n-2 β n-2,0 p cn-2+(n-2) • • • ν -1 0 p c0     
in an appropriate basis e = (e n-1 , e n-2 , • • • , e 0 ), which can be rewritten as

Mat e (φ) =      ν -1 n-1 0 • • • 0 ν -1 n-1 β n-1,n-2 ν -1 n-2 • • • 0 . . . . . . . . . . . . ν -1 n-1 β n-1,0 ν -1 n-2 β n-2,0 • • • ν -1 0      =:H •Diag p cn-1+n-1 , • • • , p c1+1 , p c0 .
By considering the change of basis e = e • H and then reversing the order of the basis e , the Frobenius φ of M 0 with respect to this new basis is described as follows:

Mat(φ) = Diag p c0 , p c1+1 , • • • , p cn-1+(n-1)      ν -1 0 ν -1 1 β 1,0 • • • ν -1 n-1 β n-1,0 0 ν -1 1 • • • ν -1 n-1 β n-1,1 . . . . . . . . . . . . 0 0 0 ν -1 n-1      =:H (3.3.28)
with respect to the new basis described as above.

The last displayed upper-triangular matrix H is the Frobenius of the Fontaine-Laffaille module M such that T * cris (M ) ∼ = ρ 0 ∼ = T r st (M), by Lemma 3.2.27. Hence, we get the desired results (c.f. Definition 3.3.15). Remark 3.3.29. We emphasize that the matrix H is the Frobenius of the Fontaine-Laffaille module M , with respect to a basis (e 0 , e 1 , • • • , e n-1 ) compatible with the filtration, such that

T * cris (M ) ∼ = ρ 0 ∼ = T r st (M),
H =      1 β 1,0 • • • β n-1,0 0 1 • • • β n-1,1 . . . . . . . . . . . . 0 0 0 1      =:H • Diag ν -1 0 , ν -1 1 , • • • , ν -1 n-1 , so that we have (1, n)w 0 H t ∈ B(F)w 0 B(F) if and only if (1, n)w 0 (H ) t ∈ B(F)w 0 B(F).
Hence, ρ 0 being Fontaine-Laffaille generic is a matter only of the entries of the filtration of the Breuil modules if the Breuil modules are written as in Lemma 3.3.21.

Fontaine-Laffaille parameters vs Frobenius eigenvalues

In this section, we study further the Breuil modules of Lemma 3.3.21. We show that if the filtration is of a certain shape then a certain product of Frobenius eigenvalues (of the Breuil modules) corresponds to the newest Fontaine-Laffaille parameter, FL n-1,0 n (ρ 0 ). To get such a shape of the filtration, we assume further that ρ 0 is Fontaine-Laffaille generic.

Lemma 3.3.30. Keep the assumptions and notation of Lemma 3.3.21. Assume further that ρ 0 is Fontaine-Laffaille generic (c.f. Definition 3.3.18).

Then M ∈ F-BrMod n-1 dd can be described as follows: there exist a framed basis e for M and a framed system of generators f for Fil n-1 M such that

Mat e,f (φ n-1 ) = Diag (µ n-1 , µ n-2 , • • • , µ 0 ) and Mat e,f (Fil n-1 M) = (U i,j )
where

U i,j =                    u r (0) n-1 e-(k (0) n-1 -k (0) 0 ) if i = n -1 and j = 0; u r (0) i e if 0 < i = j < n -1; x i,j • u r (0) i e-(k (0) i -k (0) j ) if n -1 > i > j; u r (0) 0 e+(k (0) n-1 -k (0) 0 ) if i = 0 and j = n -1; x 0,j • u r (0) 0 e+(k (0) j -k (0) 0 ) if i = 0 ≤ j < n -1; 0 otherwise. (3.3.31)
Here, µ i ∈ F × and x i,j ∈ F. Moreover, we have the following identity:

FL n-1,0 n (ρ 0 ) = n-2 i=1 µ -1 i .
Due to the size of the matrix, we decide to describe the matrix Mat e,f (Fil n-1 M) as (3.3.31). But for the reader we visualize the matrix Mat e,f (Fil n-1 M) below, although it is less accurate:

         0 0 • • • 0 u r (0) n-1 e-k (0) n-1,0 0 u r (0) n-2 • • • x n-2,1 u r (0) n-2 e-k (0) n-2,1 x n-2,0 u r (0) n-2 e-k (0) n-2,0 . . . . . . . . . . . . . . . 0 0 • • • u r (0) 1 x 1,0 u r (0) 1 e-k (0) 1,0 u r (0) 0 e+k (0) n-1,0 x 0,n-2 u r (0) 0 e+k (0) n-2,0 • • • x 0,1 u r (0) 0 e+k (0) 1,0 x 0,0 u r (0) 0 e         
where k

(0) i,j := k (0) i -k (0) j .
Proof. Let e 0 be a framed basis for M and f 0 a framed system of generators for Fil n-1 M such that V 0 := Mat e 0 ,f 0 (Fil n-1 M) and A 0 := Mat e 0 ,f 0 (φ n-1 ) are given as in Lemma 3.3.21. So, in particular, V 0 is upper-triangular and A 0 is diagonal. By Proposition 3.3.27, the upper-triangular matrix H in (3.3.28) is the Frobenius of the Fontaine-Laffaille module corresponding to ρ 0 , as in Definition 3.3.15. Since we assume that ρ 0 is Fontaine-Laffaille generic, we have (1, n)w 0 H t ∈ B(F)w 0 B(F) as discussed right after Definition 3.3.15, so that we have w 0 H t w 0 ∈ (1, n)B(F)w 0 B(F)w 0 . Equivalently, w 0 (H ) t w 0 ∈ (1, n)B(F)w 0 B(F)w 0 by Remark 3.3.29, where H is defined in Remark 3.3.29. Hence, comparing V 0 with w 0 (H ) t w 0 , there exists a lower-triangular matrix C ∈ GL n (S) such that

V 0 • C = V 1 := (U i,j ) 0≤i,j≤n-1
where U i,j is described as in (3.3.31), since any matrix in w 0 B(F)w 0 is lower-triangular. From the identity

V 0 • C = V 1 , we have V 1 = Mat e 1 ,f 1 (Fil n-1 M) and A 1 := Mat e 1 ,f 1 (φ n-1 ) = A 0 • φ(C) by Lemma 3.2.19
, where e 1 := e 0 and f

1 := e 1 V 1 . If i < j, then [k (0) j -k (0) i ] 1 = k (0) j -k (0) i ≥ n as ρ 0 is strongly generic, so that A 1 is congruent to a diagonal matrix B 2 ∈ GL n (F) modulo (u ne ) as C = (c i,j • u [k (0) j -k (0) i ]1
) is a lower-triangular and A 0 is diagonal. Let V 2 be the matrix obtained from V 1 by replacing x i,j in (3.3.31) by y i,j , and

B 2 = (b i,j ) is the diagonal matrix defined by taking b i,i = b i,i if 1 ≤ i ≤ n -2 and b i,i = b n-1-i,n-1-i otherwise, where B 2 = (b i,j
). Then it is obvious that there exist y i,j ∈ F such that

A 1 • V 2 ≡ V 1 • B 2 modulo (u ne ). Letting e 2 := e 1 • A 1 , we have V 2 = Mat e 2 ,f 2 (Fil n-1 M) and Mat e 2 ,f 2 (φ n-1 ) = φ(B 2 ) by Lemma 3.2.19. Note that A 2 := Mat e 2 ,f 2 (φ n-1
) is diagonal. Hence, there exist a framed basis for M and a framed system of generators for Fil n-1 M such that Mat e,f (φ n-1 ) and Mat e,f (Fil n-1 M) are described as in the statement.

We now prove the second part of the lemma. It is harmless to assume c 0 = 0 by Lemma 3.3.17. Let V := Mat e,f (Fil n-1 M) and A := Mat e,f (φ n-1 ) be as in the first part of the lemma. By Lemma 3.2.26, the φ-module over F ⊗ Fp F p (( )) defined by M := M Fp(( )) (M * ) is described as follows: there exists a basis e = (e n-1 , e n-2 , • • • , e 0 ), compatible with decent data, such that Mat e (φ) = ( A -1 V ) t where V t and ( A -1 ) t are computed as follows:

V t =          0 0 • • • 0 r (0) 0 e+k (0) n-1,0 0 r (0) n-2 • • • 0 x 0,n-2 r (0) 0 e+k (0) n-2,0 . . . . . . . . . . . . . . . 0 x n-2,1 r (0) n-2 e-k (0) n-2,1 • • • r (0) 1 x 0,1 r (0) 0 e+k (0) 1,0 r (0) n-1 e-k (0) n-1,0 x n-2,0 r (0) n-2 e-k (0) n-2,0 • • • x 1,0 r (0) 1 e-k (0) 1,0 x 0,0 r (0) 0 e          and
A -1 = Diag µ -1 n-1 , µ -1 n-2 , • • • , µ -1 0 .
By considering the change of basis e = ( k (0)

n-1 e n-1 , k (0) n-2 e n-2 , • • • , k (0) 1 e 1 , k (0) 0 e 0 ), we have Mat e (φ) = ( V t ) • Diag µ -1 n-1 , µ -1 n-2 , • • • , µ -1
where

( V t ) =          0 0 • • • 0 e(k (0) 0 +r (0) 0 ) 0 e(k (0) n-2 +r (0) n-2 ) • • • 0 x 0,n-2 e(k (0) 0 +r (0) 0 ) . . . . . . . . . . . . . . . 0 x n-2,1 e(k (0) n-2 +r (0) n-2 ) • • • e(k (0) 1 +r (0) 1 )
x 0,1 e(k

(0) 0 +r (0) 0 ) e(k (0)
n-1 +r

(0) n-1 )
x n-2,0 e(k

(0) n-2 +r (0) n-2 ) • • • x 1,0 e(k (0) 1 +r (0) 1 )
x 0,0 e(k

(0) 0 +r (0) 0 )          . Since k (0) j + r (0) j = c j + j for all j, it is immediate that the φ-module M over F ⊗ Fp F p (( )) is the base change via F ⊗ Fp F p ((p)) → F ⊗ Fp F p (( )) of the φ-module M 0 over F ⊗ Fp F p ((p)) described by Mat e (φ) = F • Diag p cn-1+n-1 , p cn-2+n-2 , • • • , p c0
where

F =          0 0 0 • • • 0 µ -1 0 0 µ -1 n-2 0 • • • 0 µ -1 0 x 0,n-2 0 µ -1 n-2 x n-2,n-3 µ -1 n-3 • • • 0 µ -1 0 x 0,n-3 . . . . . . . . . . . . . . . . . . 0 µ -1 n-2 x n-2,1 µ -1 n-3 x n-3,1 • • • µ -1 1 µ -1 0 x 0,1 µ -1 n-1 µ -1 n-2 x n-2,0 µ -1 n-3 x n-3,0 • • • µ -1 1 x 1,0 µ -1 0 x 0,0         
, in an appropriate basis e . Now, consider the change of basis e = e • F and then reverse the order of the basis e . Then the matrix of the Frobenius φ for M 0 with respect to this new basis is given by

Diag p c0 , p c1+1 , • • • , p cn-1+n-1 • F where F =          µ -1 0 x 0,0 µ -1 1 x 1,0 µ -1 2 x 2,0 • • • µ -1 n-2 x n-2,0 µ -1 n-1 µ -1 0 x 0,1 µ -1 1 µ -1 2 x 2,1 • • • µ -1 n-2 x n-2,1 0 µ -1 0 x 0,2 0 µ -1 2 • • • µ -1 n-2 x n-2,2 0 . . . . . . . . . . . . . . . . . . µ -1 0 x 0,n-2 0 0 • • • µ -1 n-2 0 µ -1 0 0 0 • • • 0 0         
.

By Lemma 3.2.27, there exists a Fontaine-Laffaille module M such that F(M ) = M 0 with Hodge-Tate weights (c 0 , c 1 + 1, • • • , c n-1 + n -1) and Mat e (φ • ) = F for some basis e of M compatible with the Hodge filtration on M . On the other hand, since T * cris (M ) ∼ = ρ 0 , there exists a basis e of M compatible with the Hodge filtration on M such that

Mat e (φ • ) =        w 0 w 0,1 • • • w 0,n-2 w 0,n-1 0 w 1 • • • w 1,n-2 w 1,n-1 . . . . . . . . . . . . . . . 0 0 • • • w n-2 w n-2,n-1 0 0 • • • 0 w n-1        =:G
where w i,j ∈ F and w i ∈ F × by Lemma 3.3.12. Since both e and e are compatible with the Hodge filtration on M , there exists a unipotent lower-triangular n × n-matrix U such that

U • F = G.
Note that we have w 0,n-1 = µ -1 n-1 by direct computation. Let U be the (n -1) × (n -1)-matrix obtained from U by deleting the right-most column and the lowest row, and F (resp. G ) the (n -1) × (n -1)-matrix obtained from F (resp. G) by deleting the left-most column and the lowest row. Then they still satisfy G = U • F as U is a lower-triangular unipotent matrix, so that

FL n-1,0 n (ρ 0 ) = [w 0,n-1 : (-1) n det G ] = µ -1 n-1 : (-1) n det F = 1 : n-2 i=1 µ -1 i ,
which completes the proof.

Proposition 3.3.32. Keep the assumptions and notation of Lemma 3.3.30.

Then M ∈ F-BrMod n-1 dd can be described as follows: there exist a framed basis e for M and a framed system of generators f for Fil n-1 M such that

Mat e,f (Fil n-1 M) =           0 0 0 • • • 0 u e-(k (0) n-1 -k (0) 0 ) 0 u (n-2)e 0 • • • 0 0 0 0 u (n-3)e • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 • • • u e 0 u (n-2)e+(k (0) n-1 -k (0) 0 ) 0 0 • • • 0 0           . Moreover, if we let Mat e,f (φ n-1 ) = α i,j u [k (0) j -k (0) i ]1
for α i,i ∈ S × 0 and α i,j ∈ S 0 if i = j then we have the following identity:

FL n-1,0 n (ρ 0 ) = n-2 i=1 (α (0) i,i ) -1 = n-2 i=1 µ -1 i where α (0) i,j ∈ F is determined by α (0) i,j ≡ α i,j modulo (u e ).
Note that Mat e,f (φ n-1 ) always belong to GL n (S) as e and f are framed.

Proof. We let e 0 (resp. e 1 ) be a framed basis for M and f 0 (resp. f 1 ) be a framed system of generators for Fil n-1 M such that Mat e 0 ,f 0 (Fil n-1 M) and Mat e 0 ,f

0 (φ n-1 ) (resp. Mat e 1 ,f 1 (Fil n-1 M) and Mat e 1 ,f 1 (φ n-1
)) are given as in the statement of Lemma 3.3.30 (resp. in the statement of Proposition 3.3.32). We also let V 0 = Mat e 0 ,f 0 (Fil n-1 M) and A 0 = Mat e 0 ,f 0 (φ n-1 ) as well as

V 1 = Mat e 1 ,f 1 (Fil n-1 )M and A 1 = Mat e 1 ,f 1 (φ n-1 ). It is obvious that there exist R = (r i,j u [k (0) j -k (0) i ]1 ) and C = (c i,j u [k (0) j -k (0) i ]1 ) in GL n (S) such that R • V 0 • C = V 1 and e 1 = e 0 R -1
for r i,j and c i,j in S 0 . From the first equation above, we immediately get the identities:

r (0) n-1,n-1 • c (0) 0,0 = 1 = r (0) 0,0 • c (0) n-1,n-1 and r (0) i,i • c (0) i,i = 1 for 0 < i < n -1, where r (0) i,j ∈ F (resp. c (0) i,j ∈ F) is determined by r (0)
i,j ≡ r i,j modulo (u e ) (resp. c (0) i,j ≡ c i,j modulo (u e )). By Lemma 3.2.19, we see that

A 1 = R • A 0 • φ(C). 64CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Hence, if we let A 1 = α i,j u [k (0) j -k (0) i ]1 then r (0) i,i • µ i • c (0) i,i = α (0) i,i
for each 0 < i < n -1 since R and C are diagonal modulo (u), so that we have

n-2 i=1 µ i = n-2 i=1 α (0) i,i
which completes its proof.

Note that the matrix in the statement of Proposition 3.3.32 gives rise to the elementary divisors of M/Fil n-1 M.

Filtration of strongly divisible modules

In this section, we describe the filtration of the strongly divisible modules lifting the Breuil modules described in Proposition 3.3.32. Throughout this section, we keep the notation r (0) i as in (3.3.20) as well as k (0) i . We start to recall the following lemma, which is easy to prove but very useful. Lemma 3.3.33. Let 0 < f ≤ n be an integer, and let M ∈ O E -Mod n-1 dd be a strongly divisible module corresponding to a lattice in a potentially semi-stable representation ρ :

G Qp → GL n (E) with Hodge- Tate weights {-(n-1), -(n-2), • • • , 0} and Galois type of niveau f such that T Qp,n-1 st ( M)⊗ O E F ∼ = ρ 0 . If we let X (i) := Fil n-1 M ∩ Fil i S • M Fil n-1 S • M ⊗ O E E for i ∈ {0, 1, • • • , n -1}, then for any character ξ : Gal(K/K 0 ) → K × we have that the ξ-isotypical component X (i) ξ of X (i) is a free K 0 ⊗ E-module of finite rank rank K0⊗ Qp E X (i) ξ = n(n -1) 2 - i(i + 1) 2 .
Moreover, multiplication by u ∈ S induces an isomorphism

X (0) ξ ∼ -→ X (0) ξ ω .
Proof. Since ρ has Hodge-Tate weights {-(n-1), -(n-2), • • • , 0}, by the analogue with E-coefficients of [Bre97], Proposition A.4, we deduce that

Fil n-1 D = Fil n-1 S E f n-1 ⊕ Fil n-2 S E f n-2 ⊕ • • • ⊕ Fil 1 S E f 1 ⊕ S E f 0 for some S E -basis f 0 , • • • , f n-1 of D, where D := M[ 1 p ] ∼ = S E ⊗ E D Qp,n-1 st (V )
, so that we also have

Fil n-1 D ∩ Fil i S E D = Fil n-1 S E f n-1 ⊕ Fil n-2 S E f n-2 ⊕ • • • ⊕ Fil i S E f i ⊕ • • • ⊕ Fil i S E f 0 . Since ρ ∼ = T Qp,n-1 st ( M) ⊗ O E E is a G Qp -representation, Fil i (K ⊗ K0 D Qp,n-1 st (ρ)) ∼ = K ⊗ Qp Fil i D dR (ρ ⊗ ε 1-n ), so that X (i) ∼ = Fil n-1 D∩Fil i S E D Fil n-1 S E D is a free K 0 ⊗ Qp E-module. Since S E Fil n-1 S E ∼ = n-2 i=0 e-1 j=0 (K 0 ⊗ Qp E)u j E(u) i , we have rank K0⊗ Qp E X (i) = n(n-1) 2 -i(i+1)
2 e. We note that Gal(K/K 0 ) acts semisimply and that multiplication by u gives rise to an K 0 ⊗ Qp E-linear isomorphism on S E /Fil p S E which cyclically permutes the isotypical components, which completes the proof.

Note that Lemma 3.3.33 immediately implies that

rank K0⊗ Qp E X (i) ξ -rank K0⊗ Qp E X (i+1) ξ = i + 1.
(3.3.34)

We will use this fact frequently to prove the main result, Proposition 3.3.36, in this subsection.

To describe the filtration of strongly divisible modules, we need to analyze the Fil n-1 M of the Breuil modules M we consider.

Lemma 3.3.35. Keep the notation and assumptions of Lemma 3.3.21.

(i) If u a is an elementary divisor of M/Fil n-1 M then e -(k (0) n-1 -k (0) 0 ) ≤ a ≤ (n -2)e + (k (0) n-1 -k (0) 0 ). Moreover, FL n-1,0 n (ρ 0 ) = ∞ (resp. FL n-1,0 n (ρ 0 ) = 0) if and only if u e-(k (0) n-1 -k (0) 0 ) (resp. u (n-2)e+(k (0) n-1 -k (0) 0 ) ) is an elementary divisor of M/Fil n-1 M.
(ii) If we further assume that ρ 0 is Fontaine-Laffaille generic, then

{u (n-2)e+(k (0) n-1 -k (0) 0 ) , u (n-2)e , u (n-3)e , • • • , u e , u e-(k (0) n-1 -k (0) 0 ) } are the elementary divisors of M/Fil n-1 M.
Proof. The first part of (i) is obvious since one can obtain the Smith normal form of Mat e,f Fil n-1 M by elementary row and column operations. By Proposition 3.3.27, we know that FL n-1,0 n

(ρ 0 ) = ∞ if and only if β n-1,0 = 0. Since u e-(k (0) n-1 -k (0)
0 ) has the minimal degree among the entries of Mat e,f Fil n-1 M, we conclude the equivalence statement for FL n-1,0 n (ρ 0 ) = ∞ holds. The last part of (i) is immediate from the other equivalence statement, FL n-1,0 n (ρ 0 ) = ∞ if and only if β n-1,0 = 0, by considering M * and using Lemma 3.3.17, (vi).

Part (ii) is obvious from Proposition 3.3.32.

Proposition 3.3.36. Assume that ρ 0 is Fontaine-Laffaille generic and keep the notation r

-(n -1), -(n -2), • • • , 0} such that T Qp,n-1 st ( M) ⊗ O E F ∼ = ρ 0 .
Then there exists a framed basis ( e n-1 , e n-2 , • • • , e 0 ) for M and a framed system of generators

( f n-1 , f n-2 , • • • , f 0 ) for Fil n-1 M modulo Fil n-1 S • M such that Mat e, f Fil n-1 M is described as fol- lows:           -p n-1 α 0 0 • • • 0 u e-(k (0) n-1 -k (0) 0 ) 0 E(u) n-2 0 • • • 0 0 0 0 E(u) n-3 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 • • • E(u) 0 u k (0) n-1 -k (0) 0 n-2 i=0 p n-2-i E(u) i 0 0 • • • 0 α           where α ∈ O E with 0 < v p (α) < n -1.
Proof. Note that we write the elements of M in terms of coordinates with respect to a framed basis e := ( e n-1 , e n-2 , • • • , e 0 ). We let M := M ⊗ S S, which is a Breuil module of weight n -1 and of type

n-1 i=0 ω k (0)
i by Proposition 3.2.18. Note also that M can be described as in Proposition 3.3.32, and we assume that M has such a framed basis for M and such a framed system of generators for Fil n-1 M. Let

f 0 =         u e-(k (0) n-1 -k (0) 0 ) n-2 k=0 x n-1,k E(u) k u e-(k (0) n-2 -k (0) 0 ) n-2 k=0 x n-2,k E(u) k . . . u e-(k (0) 1 -k (0) 0 ) n-2 k=0 x 1,k E(u) k n-2 k=0 x 0,k E(u) k         ∈ Fil n-1 M Fil n-1 S M ω k (0) 0 ,
where x i,j ∈ O E . The vector f 0 can be written as follows:

f 0 = u e-(k (0) n-1 -k (0) 0 )          n-2 k=0 x n-1,k E(u) k u (k (0) n-1 -k (0) n-2 ) n-2 k=0 x n-2,k E(u) k . . . u (k (0) n-1 -k (0) 1 ) n-2 k=0 x 1,k E(u) k u (k (0) n-1 -k (0) 0 ) n-2 k=1 x 0,k [E(u) k -p k ]/u e          =: e n-1 +        0 0 . . . 0 x 0,0 + n-2 k=1 x 0,k p k        . By (ii) of Lemma 3.3.35, we know that u e-(k (0) n-1 -k (0) 0 )
is an elementary divisor of M/Fil n-1 M and all other elementary divisors have bigger powers, so that we may assume v p (x n-1,0 ) = 0. Since

Fil n-1 M ⊆ u e-(k (0) n-1 -k (0) 0 ) M, we must have v p (x 0,0 ) > 0. So e 1 := ( e n-1 , e n-2 , • • • , e 0 )
is a framed basis for M by Nakayama lemma and we have the following coordinates of f 0 with respect to e 1 :

f 0 =        u e-(k (0) n-1 -k (0) 0 ) 0 . . . 0 α        ∈ Fil n-1 M Fil n-1 S M ω k (0) 0 for α ∈ O E with v p (α) > 0. Since u k (0) 1 -k (0) 0 f 0 ∈ Fil n-1 M Fil n-1 S• M ω k (0) 1
, there exists f 1 such that

f 1 =         0 u e-(k (0) n-2 -k (0) 1 ) n-2 k=0 y n-2,k E(u) k . . . n-2 k=0 y 1,k E(u) k u k (0) 1 -k (0) 0 n-2 k=0 y 0,k E(u) k         ∈ Fil n-1 M Fil n-1 S M ω k (0) 1 ,
where y i,j ∈ O E . By Lemma 3.3.33, we have y i,0 = 0 for all i: otherwise, both

u k (0) 1 -k (0) 0 f 0 and f 1 belong to X (0) ω k (0) 1 -X (1) ω k (0)
1 which violates (3.3.34). Since u e is an elementary divisor of M/Fil n-1 M by (ii) of Lemma 3.3.35, we may also assume y 1,1 = 1. Hence, by the obvious change of basis we get f 1 as follows:

f 1 = E(u)        0 . . . 0 1 0        ∈ Fil n-1 M Fil n-1 S M ω k (0) 1 .
By the same arguments, we get

f i ∈ Fil n-1 M Fil n-1 S M ω k (0) i for i = 1, 2, • • • , n -2
as in the statement. Note that the elements in the set

{u k (0) n-1 -k (0) 0 f 0 , E(u)u k (0) n-1 -k (0) 0 f 0 , • • • , E(u) n-2 u k (0) n-1 -k (0) 0 f 0 } ∪ {u k (0) n-1 -k (0) 1 f 1 , E(u)u k (0) n-1 -k (0) 1 f 1 , • • • , E(u) n-3 u k (0) n-1 -k (0) 1 f 1 } ∪ • • • ∪ {u k (0) n-1 -k (0) n-2 f n-2 } are linearly independent in X (0) ω k (0) n-1
over E, so that the set forms a basis for X (0)

ω k (0) n-1
by Lemma 3.3.33.

Hence, f n-1 is a linear combination of those elements over E. We have

u k (0) n-1 -k (0) 0 n-2 i=0 p n-2-i E(u) i f 0 =        -p n-1 0 . . . 0 αu k (0) n-1 -k (0) 0 n-2 i=0 p n-2-i E(u) i       
.

Hence, we may let

f n-1 := 1 α u k (0) n-1 -k (0) 0 n-2 i=0 p n-2-i E(u) i f 0 ∈ Fil n-1 M Fil n-1 S M ω k (0) n-1 since u (n-2)e+(k (0) n-1 -k (0) 0 ) is an elementary divisor for M/Fil n-1 M by (ii) of Lemma 3.3.35. Moreover, v p p n-1 α > 0 since Fil n-1 M ⊆ u e-(k (0) n-1 -k (0) 0 ) M ⊆ uM by Proposition 3.3.32.
It is obvious that the f i mod ( E , Fil p S) generate M/Fil n-1 M for M written as in Proposition 3.3.27. By Nakayama Lemma, we conclude that the f i generate M/Fil n-1 M, which completes the proof.

Corollary 3.3.37. Keep the notation and assumptions of Proposition 3.3.36, and let

(λ n-1 , λ n-2 , • • • , λ 0 ) ∈ (O E ) n be the Frobenius eigenvalues on the ( ω k (0) n-1 , ω k (0) n-2 , • • • , ω k (0) 0 )-isotypic component of D Qp,n-1 st (ρ). Then v p (λ i ) =    v p (α) if i = n -1 (n -1) -i if n -1 > i > 0 (n -1) -v p (α) if i = 0.
Proof. The proof goes parallel to the proof of [HLM17], Corollary 2.4.11.

Reducibility of certain lifts

In this section, we let 1 ≤ f ≤ n and e = p f -1, and we prove that every potentially semi-stable lift of ρ 0 with Hodge-Tate weights {-(n -1), -(n -2), • • • , 0} and certain prescribed Galois types n-1 i=0 ω ki f is reducible. We emphasize that we only assume that ρ 0 is generic (c.f. Definition 3.3.3) for the results in this section.

Proposition 3.3.38. Assume that ρ 0 is generic, and let (k n-1 , k n-2 , • • • , k 0 ) be an n-tuple of integers. Assume further that k 0 ≡ (p f -1 + p f -2 + • • • + p + 1)c 0 modulo (e) and that k i are pairwise distinct modulo (e). f is an extension of a 1-dimensional potentially semi-stable lift of ρ 0,0 with Hodge-Tate weight 0 and Galois type ω k0 f by an (n -1)-dimensional potentially semi-stable lift of ρ n-1,1 with Hodge-Tate weights {-(n -1), -(n -2), • • • , 1} and Galois types n-1 i=1 ω ki f . Note that if f = 1 then the assumption that ρ 0 is generic implies that k i are pairwise distinct modulo (e) by Lemma 3.3.10. In fact, we believe that this is true for any 1 ≤ f ≤ n, but this requires extra works as we did in Lemma 3.3.10. Since we will need the results in this section only when f = 1, we will add the assumption that k i are pairwise distinct modulo (e) in the proposition.

Proof. Let M ∈ O E -Mod n-1
dd be a strongly divisible module corresponding to a Galois stable lattice in a potentially semi-stable representation ρ :

G Qp → GL n (E) with Galois type n-1 i=0 ω ki f and Hodge- Tate weights {-(n -1), -(n -2), • • • , 0} such that T Qp,n-1 st ( M) ⊗ O E F ∼ = ρ 0 .
We also let M be the Breuil module corresponding to the mod p reduction of M. M (resp. M) is of inertial type

n-1 i=0 ω ki f (resp. n-1 i=0 ω ki f ) by Proposition 3.2.18. We let f = (f n-1 , f n-2 , • • • , f 0 ) (resp. f = ( f n-1 , f n-2 , • • • , f 0 )
) be a framed system of generators for Fil n-1 M (resp. for Fil n-1 M). We also let e = (e n-1 , e n-2 , • • • , e 0 ) (resp. e = ( e n-1 , e n-2 , • • • , e 0 )) be a framed basis for M (resp. for M). If x = a n-1 e n-1 + • • • + a 0 e 0 ∈ M, we will write [x] ei for a i for i ∈ {0, 1, • • • , n -1}. We define [x] ei for x ∈ M in the obvious similar way. We may assume that Mat e,f (Fil n-1 M), Mat e,f (φ n-1 ), and Mat e (N ) are written as in (3.3.5), (3.3.6), and (3.3.7) respectively, and we do so.

By the equation (3.3.4), we deduce r 0 ≡ 0 modulo (e) from our assumption on k 0 . Recall that p > n 2 + 2(n -3) by the generic condition. Since 0 ≤ r 0 ≤ (n -1)(p f -1)/(p -1) by (ii) of Lemma 3.2.10, we conclude that r 0 = 0. Thus, we may let f 0 satisfy that [f 0 ] ei = 0 if 0 < i ≤ n -1 and [f 0 ] e0 = 1, so that we can also let

f 0 =      0 . . . 0 1      .
Hence, we can also assume that [ f j ] e0 = 0 for 0 < j ≤ n -1. We let

V 0 = Mat e, f (Fil n-1 M) ∈ M , n (S O E ) and A 0 = Mat e, f (φ n-1 ) ∈ GL n (S O E ).
We construct a sequence of framed bases { e (m) } for M by change of basis, satisfying that

Mat e (m) , f (m) (Fil n-1 M) ∈ M , n (S O E ) and Mat e (m) , f (m) (φ n-1 ) ∈ GL n (S O E )
converge to certain desired forms as m goes to ∞. We let V (m) ∈ M , n (S O E ) and A (m) ∈ GL n (S O E ) for a non-negative integer m. We may write (x

(m+1) n-1 u [kn-1-k0] f , x (m+1) n-2 u [kn-2-k0] f , • • • , x (1) m+1 u [km+1-k0] f , x (m+1) 0 )
for the last row of (A (m) ) -1 , where x

(m+1) 0 ∈ (S × O E ) 0 and x (m+1) j ∈ (S O E ) 0 for 0 < j ≤ n -1.
We define an n × n-matrix R (m+1) as follows:

R (m+1) =         1 0 • • • 0 0 0 1 • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • 1 0 x (m+1) n-1 x (m+1) 0 u [kn-1-k0] f x (m+1) n-2 x (m+1) 0 u [kn-2-k0] f • • • x (m+1) 1 x (m+1) 0 u [k1-k0] f 1         .
We also define

C (m+1) =        1 0 • • • 0 0 0 1 • • • 0 0 . . . . . . . . . . . . . . . 0 0 • • • 1 0 y (m+1) n-1 u [p -1 (kn-1-k0)] f y (m+1) n-2 u [p -1 (kn-2-k0)] f • • • y (m+1) 1 u [p -1 (k1-k0)] f 1        by the equation R (m+1) • V (m) • C (m+1) = V (m)
where y

(m+1) j ∈ (S O E ) 0 for 0 < j ≤ n -1.
Note that the existence of such a matrix C (m+1) is obvious, since p -1 k 0 ≡ k 0 modulo (e) by our assumption on k 0 immediately implies

[p -1 (k j -k 0 )] f ≤ [k s -k 0 ] f + [p -1 k j -k s ] f . We also note that R (m+1) ∈ GL n (S O E ) and C (m+1) ∈ GL , n (S O E ). Let V (m+1) = V (m) for all m ≥ 0. Assume that V (m) = Mat e (m) , f (m) (Fil n-1 M) and A (m) =
Mat e (m) , f (m) (φ n-1 ), with respect to a framed basis e (m) and a framed system of generators f

(m) . If we let e (m+1) = e (m) • (R (m+1) ) -1 , then φ n-1 ( e (m+1) V (m+1) ) = φ n-1 ( e (m) (R (m+1) ) -1 V (m+1) ) = φ n-1 ( e (m) V (m) C (m+1) ) = e (m) A (m) φ(C (m+1) ) = e (m+1) R (m+1) • A (m) • φ(C (m+1) ).
Hence, we get

V (m+1) = Mat e (m+1) , f (m+1) (Fil n-1 M) and R (m+1) • A (m) • φ(C (m+1) ) = Mat e (m+1) , f (m+1) (φ n-1 ),
where f

:= e (m+1) V (m+1) . We compute the matrix product A (m+1) := R (m+1) • A (m) • φ(C (m+1) ) as it follows. If we let

A (m) = α (m) i,j u [kj -ki] f 0≤i,j≤n-1 for α (m) i,j ∈ (S O E ) 0 if i = j and α (m) i,i ∈ (S × O E ) 0 , then A (m+1) = α (m+1) i,j u [kj -ki] f 0≤i,j≤n-1 ∈ GL n (S O E ) (3.3.39)
where α (m+1) i,j u [kj -ki] f is described as follows:

           α (m) i,j u [kj -ki] f + α (m) i,0 u [k0-ki] f φ(y (m+1) j )u p[p -1 (kj -k0)] f if i > 0 and j > 0; α (m) i,0 u [k0-ki] f if i > 0 and j = 0; 1 x (m+1) 0 φ(y (m+1) j )u p[p -1 (kj -k0)] f if i = 0 and j > 0; 1 x (m+1) 0 if i = 0 and j = 0.
Let V (0) = V 0 and A (0) = A 0 . We apply the algorithm above to V (0) and A (0) . By the algorithm above, we have two matrices V (m) and A (m) for each m ≥ 0. We claim that

         α (m+1) i,j -α (m) i,j ∈ u (1+p+•••+p m )e S O E if i > 0 and j > 0; α (m+1) i,j = α (m) i,j if i > 0 and j = 0; α (m+1) i,j ∈ u (1+p+•••+p m )e S O E if i = 0 and j > 0; α (m+1) i,j -α (m) i,j ∈ u (1+p+•••+p m-1 )e S O E if i = 0 and j = 0. 70CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
It is obvious that the case i > 0 and j = 0 from the computation (3.3.39). For the case i = 0 and j > 0 we induct on m. Note that p

[p -1 (k j -k 0 )] f -[k j -k 0 ] f = p([p -1 k j ] f -k 0 ) -(k j -k 0 ) ≥ e if j > 0.
From the computation (3.3.39) again, it is obvious that it is true for m = 0. Assume that it holds for m. This implies that x (m+1) j ∈ u (1+p+•••+p m-1 )e S O E for 0 < j ≤ n -1 and so

y (m+1) j ∈ u (1+p+•••+p m-1 )e S O E . Since φ(y (m+1) j )u p[p -1 (kj -k0)] f -[kj -k0] f ) ∈ u (1+p+•••+p m )e S O E
, by the computation (3.3.39) we conclude that the case i = 0 and j > 0 holds. The case i > 0 and j > 0 follows easily from the case i = 0 and j > 0, since [p -1 (k j -k

0 )] f + [k 0 -k i ] f -[k j -k i ] f = p([p -1 k j ] f -k 0 ) + e + k 0 -k i -[k j -k i ] f ≥ p[p -1 k j ] f -k j -(p -1)k 0 ≥ e.
Finally, we check the case i = 0 and j = 0. We also induct on m for this case. It is obvious that it holds for m = 0. Note that

R (m+1) ≡ I n modulo u (1+p+•••+p m-1 )e S O E . Since A (m+1) = R (m+1) • A (m) • φ(C (m+1)
), we conclude that the case i = 0 and j = 0 holds. The previous claim says the limit of A (m) exists (entrywise), say A (∞) . By definition, we have V (∞) = V (m) for all m ≥ 0. In other words, there exist a framed basis e (∞) for M and a framed system of generators f

(∞) for Fil n-1 M such that Mat e (∞) , f (∞) (Fil n-1 M) = V (∞) ∈ M , n (S O E ) and Mat e (∞) , f (∞) (φ n-1 ) = A (∞) ∈ GL n (S O E ).
Note that (V (∞) ) i,j = 0 if either i = 0 and j > 0 or i > 0 and j = 0, and that (A (∞) ) i,j = 0 if i = 0 and j > 0. Since e (∞) is a framed basis for M, we may write

Mat e (∞) (N ) = γ i,j u [kj -ki] f 0≤i,j≤n-1 ∈ M n (S O E )
for the matrix of the monodromy operator of M where γ i,j ∈ (S O E ) 0 , and let

A (∞) = α (∞) i,j u [kj -ki] f 0≤i,j≤n-1 ∈ GL n (S O E ).
We claim that γ 0,j = 0 for n -1 ≥ j > 0. Recall that α (∞) 0,j = 0 for j > 0, and write f

(∞) = ( f (∞) n-1 , f (∞) n-2 , • • • , f (∞) 0
) and e (∞) = ( e

(∞) n-1 , e (∞) n-2 , • • • , e (∞) 0
). We also write

f (∞) j = n-1 i=1 β (∞) i,j u [p -1 kj -ki] e (∞) i where β (∞) i,j ∈ (S O E ) 0 , for each 0 < j ≤ n -1. From the equation [cN φ n-1 ( f (∞) j )] e (∞) 0 = [φ n-1 (E(u)N ( f (∞) j
))] e (∞) 0

for n -1 ≥ j > 0, we have the identity

n-1 i=1 α (∞) i,j u [kj -ki] f +[ki-k0] f γ 0,i = p n-1 i=1 β (∞) i,j u p[p -1 kj -ki] f +p[ki-k0] f φ(γ 0,i )α (∞) 0,0 (3.3.40)
for each n -1 ≥ j > 0. Choose an integer s such that ord u (γ 0,s u [ks-k0] f ) ≤ ord u (γ 0,i u [ki-k0] f ) for all n -1 ≥ i > 0, and consider the identity (3.3.40) for j = s. Then the identity (3.3.40) induces

α (∞) s,s u [ks-k0] f γ 0,s ≡ 0 modulo (u ordu(γ0,s)+[ks-k0] f +1 ). Note that α (∞)
s,s ∈ S × O E , so that we get γ 0,s = 0. Recursively, we conclude that γ 0,j = 0 for all 0 < j ≤ n -1.

Finally, it is now easy to check that ( e

(∞) n-1 , e (∞) n-2 , • • • , e (∞) 1
) determines a strongly divisible modules of rank n -1, that is a submodule of M. This completes the proof.

Corollary 3.3.41. Fix a pair of integers (i 0 , j 0 ) with 0 ≤ j 0 ≤ i 0 ≤ n -1. Assume that ρ 0 is generic, and let (k n-1 , k n-2 , • • • , k 0 ) be an n-tuple of integers. Assume further that

k i = (p f -1 + p f -2 + • • • + p + 1)c i
for i > i 0 and for i < j 0 and that the k i are pairwise distinct modulo (e).

Then every potentially semi-stable lift ρ of ρ 0 with Hodge-Tate weights {-(n -1),

-(n -2), • • • , 0} and Galois types n-1 i=0 ω ki f is a successive extension ρ ∼ =             ρ n-1,n-1 • • • * * * • • • * . . . . . . . . . . . . . . . . . . ρ i0+1,i0+1 * * • • • * ρ i0,j0 * • • • * ρ j0-1,j0-1 • • • * . . . . . . ρ 0,0             where • ρ i,i is a 1-dimensional potentially semi-stable lift of ρ i,i
with Hodge-Tate weights -i and Galois type ω k i 0 ,j 0 i for n -1 ≥ i > i 0 and for j 0 > i ≥ 0;

• ρ i0,j0 is a (i 0 -j 0 + 1)-dimensional potentially semi-stable lift of ρ i0,j0 with Hodge-Tate weights

{-i 0 , -i 0 + 1, • • • , -j 0 } and Galois types i0 i=j0 ω k i 0 ,j 0 i .
Proof. Proposition 3.3.38 implies this corollary recursively. Let M ∈ F-BrMod n-1 dd be a Breuil module corresponding to the mod p reduction of a strongly divisible module M ∈ O E -Mod n-1 dd corresponding to a Galois stable lattice in a potentially semi-stable representation ρ :

G Qp → GL n (E) with Galois type n-1 i=0 ω ki f and Hodge-Tate weights {-(n-1), -(n-2), • • • , 0} such that T Qp,n-1 st ( M)⊗ O E F ∼ = ρ 0 . Both M (resp. M) is of inertial type n-1 i=0 ω ki f (resp.
n-1 i=0 ω ki f ) by Proposition 3.2.18. We may assume that Mat e,f (Fil n-1 M), Mat e,f (φ n-1 ), and Mat e (N ) are written as in (3.3.5), (3.3.6), and (3.3.7) respectively, and we do so.

By the equation (3.3.4), it is easy to see that r i = (p f -1 +p f -2 +• • •+p+1)i for i > i 0 and for i < j 0 , by our assumption on k i . By Proposition 3.3.38, there exists an (n -1)-dimensional subrepresentation ρ n-1,1 of ρ whose quotient is ρ 0,0 which is a potentially semi-stable lift of ρ 0,0 with Hodge-Tate weight 0 and Galois type ω k0 f . Now consider ρ n-1,1 ⊗ ε -1 . Apply Proposition 3.3.38 to ρ n-1,1 ⊗ ε -1 . Recursively, one can readily check that ρ has subquotients ρ i,i for 0 ≤ i ≤ j 0 -1. Considering ρ ∨ ⊗ ε n-1 , one can also readily check that ρ has subquotients ρ i,i lifting ρ i,i for n -1 ≥ i ≥ i 0 + 1.

The results in Corollary 3.3.41 reduce many of our computations for the main results on the Galois side.

Main results on the Galois side

In this section, we state and prove the main local results on the Galois side, that connects the Fontaine-Laffaille parameters of ρ 0 with the Frobenius eigenvalues of certain potentially semi-stable lifts of ρ 0 .

Throughout this section, we assume that ρ 0 is Fontaine-Laffaille generic. We also fix f = 1 and e = p -1.

Fix i 0 , j 0 ∈ Z with 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1, and define the n-tuple of integers (r i0,j0 n-1 , r i0,j0 n-2 , • • • , r i0,j0 0 ) as follows:

r i0,j0 i :=    i if i 0 = i = j 0 ; j 0 + 1 if i = i 0 ; i 0 -1 if i = j 0 . (3.3.42)
We note that if we replace n by i 0 -j 0 + 1 in the definition of r (0) i in (3.3.20) then we have the identities:

r i0,j0 j0+i = j 0 + r (0) i (3.3.43) for all 0 ≤ i ≤ i 0 -j 0 . In particular, r n-1,0 i = r (0) i for all 0 ≤ i ≤ n -1. From the equation k i0,j0 i ≡ c i + i -r i0,j0
i mod (e) (c.f. Lemma 3.3.10, (i)), this tuple immediately determines an n-tuple

(k i0,j0 n-1 , k i0,j0 n-2 , • • • , k i0,j0 0
) of integers mod (e), which will determine the Galois types of our representations. We set

k i0,j0 i := c i + i -r i0,j0 i for all i ∈ {0, 1, • • • , n -1}.
The following is the main result on the Galois side.

Theorem 3.3.44. Let i 0 , j 0 be integers with 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1. Assume that ρ 0 is generic and that ρ i0,j0 is Fontaine-Laffaille generic. Let

(λ i0,j0 n-1 , λ i0,j0 n-2 , • • • , λ i0,j0 0 ) ∈ (O E ) n be the Frobenius eigenvalues on the ( ω k i 0 ,j 0 n-1 , ω k i 0 ,j 0 n-2 , • • • , ω k i 0 ,j 0 0 )-isotypic components of D Qp,n-1 st (ρ 0 )
where ρ 0 is a potentially semi-stable lift of ρ 0 with Hodge-Tate weights {-(n -1), -(n -2), • • • , -1, 0} and Galois types n-1 i=0 ω k i 0 ,j 0 i . Then the Fontaine-Laffaille parameter FL i0,j0 n associated to ρ 0 is computed as follows:

FL i0,j0 n (ρ 0 ) = p [(n-1)-i 0 +j 0 2 ](i0-j0-1) i0-1 i=j0+1 λ i0,j0 i ∈ P 1 (F).
We first prove Theorem 3.3.44 for the case (i 0 , j 0 ) = (n -1, 0) in the following proposition, which is the key first step to prove Theorem 3.3.44. Proposition 3.3.45. Keep the assumptions and notation of Theorem 3.3.44, and assume further (i 0 , j 0 ) = (n -1, 0). Then Theorem 3.3.44 holds.

Recall that (k

n-1,0 n-1 , • • • , k n-1,0 0
) in Proposition 3.3.45 is the same as (k

(0) n-1 , • • • , k (0)
0 ) in (3.3.20). To lighten the notation, we let k i = k n-1,0 i and λ i = λ n-1,0 i during the proof of Proposition 3.3.45. We heavily use the results in Sections 3.3.3, 3.3.4 and 3.3.5 to prove this proposition.

Proof. Let M ∈ O E -Mod n-1
dd be a strongly divisible module corresponding to a Galois stable lattice in a potentially semi-stable representation ρ 0 : G Qp → GL n (E) with Galois type n-1 i=0 ω ki and Hodge-Tate weights {-

(n -1), -(n -2), • • • , 0} such that T Qp,n-1 st ( M) ⊗ O E F ∼ = ρ 0 .
We also let M be the Breuil module corresponding to the mod p reduction of M. M (resp. M) is of inertial type n-1 i=0 ω ki (resp.

n-1 i=0 ω ki ) by Proposition 3.2.18. We let f = ( f n-1 , f n-2 , • • • , f 1 , f 0 ) be a framed system of generators for Fil n-1 M, and e = ( e n-1 , e n-2 , • • • , e 1 , e 0 ) be a framed basis for M. We may assume that Mat e, f (Fil n-1 M) is described as in Proposition 3.3.36, and we do so.

Define α i ∈ F × by the condition φ n-1 ( f i ) ≡ α i e i modulo ( E , u) for all i ∈ {0, 1, • • • , n -1}.
There exists a framed basis e = (e n-1 , e n-2 , • • • , e 0 ) for M and a framed system of generators f = (f n-1 , f n-2 , • • • , f 0 ) for Fil n-1 M such that Mat e,f (Fil n-1 M) is given as in Proposition 3.3.32 and Mat e,f (φ n-1 ) = α i,j u [kj -ki]1 ∈ GL n (S) for some α i,j ∈ S 0 with α i,i ≡ α i mod (u e ).

Recall that

f i = E(u) i e i for i ∈ {1, 2, • • • , n -2} by Proposition 3.3.36. Write φ n-1 ( f j ) = n-1 i=0 α i,j u [kj -ki
]1 e i for some α i,j ∈ S 0 . Then we get

s 0 ( α i,i ) ≡ p i λ i p n-1 (mod E ) for i ∈ {1, 2, • • • , n -2} since φ n-1 = 1 p n-1 φ for the Frobenius φ on D Qp,n-1 st (ρ 0 ), so that we have n-2 i=1 α i ≡ n-2 i=1 λ i p n-1-i (mod E ).
(Note that λi p n-1-i ∈ O × E by Corollary 3.3.37.) This completes the proof, by applying the results in Proposition 3.3.32.

We now prove Theorem 3.3.44, which is the main result on the Galois side.

Proof of Theorem 3.3.44. Recall from the identities in (3.3.43) that

(r i0,j0 i0 , r i0,j0 i0-1 , • • • , r i0,j0 j0 ) = j 0 + (1, n -2, n -3, • • • , 1, n -2)
where n := i 0 -j 0 + 1. Recall also that ρ 0 has a subquotient ρ i0,j0 that is a potentially semi-stable lift of ρ i0,j0 with Hodge-Tate weights {-i 0 , -(i 0 -1), ..., -j 0 } and of Galois type It is immediate that ρ i0,j0 := ρ i0,j0 ε -j0 ω j0 is another potentially semi-stable lift of ρ i0,j0 with Hodge-Tate weights {-(i 0 -j 0 ), -(i 0 -j 0 -1), ..., 0} and of Galois type

i0 i=j0 ω k i 0 ,j 0 i +j0 . We let (η i0 , η i0-1 , • • • , η j0 ) ∈ (O E ) i0-j0+1 (resp. (δ i0 , δ i0-1 , • • • , δ j0 ) ∈ (O E ) i0-j0+1
) be the Frobenius eigenvalues on the (

ω k i 0 ,j 0 i 0 , ω k i 0 ,j 0 i 0 -1 , • • • , ω k i 0 ,j 0 j 0 )-isotypic component of D Qp,i0-j0 st (ρ i0,j0 ) (resp. on the ( ω k i 0 ,j 0 i 0 +j0 , ω k i 0 ,j 0 i 0 -1 +j0 , • • • , ω k i 0 ,j 0 j 0 +j0 )-isotypic component of D Qp,i0-j0 st (ρ i0,j0
)). Then we have p -j0 δ i = η i for all i ∈ {j 0 , j 0 + 1, • • • , i 0 } and, by Proposition 3.3.45,

FL i0-j0,0 i0-j0+1 (ρ i0,j0 ) =     i0-1 i=j0+1 δ i   : p (i 0 -j 0 )(i 0 -j 0 -1) 2   ∈ P 1 (F).
But we also have that

p n-1-(i0-j0) η i = λ i0,j0
i for all i ∈ {j 0 , j 0 + 1, • • • , i 0 } by Corollary 3.3.41. Hence, we have

δ i = p -(n-1-i0) λ i0,j0 i for all i ∈ {j 0 , j 0 + 1, • • • , i 0 } and we conclude that FL i0,j0 n (ρ 0 ) = FL i0-j0,0 i0-j0+1 (ρ i0,j0 ) =     i0-1 i=j0+1 λ i0,j0 i   : p [(n-1)-i 0 +j 0 2 ](i0-j0-1)   ∈ P 1 (F).
(Note that FL i0,j0 n (ρ 0 ) = FL i0-j0,0 i0-j0+1 (ρ i0,j0 ) by Lemma 3.3.17.)

In the following corollary, we prove that the Weil-Deligne representation WD(ρ 0 ) associated to ρ 0 still contains Fontaine-Laffaille parameters. As we will see later, we will transport this information to the automorphic side via local Langlands correspondence.

Corollary 3.3.46. Keep the assumptions and notation of Theorem 3.3.44.

Then ρ 0 is, in fact, potentially crystalline and

WD(ρ 0 ) F-ss = WD(ρ 0 ) ∼ = n-1 i=0 Ω i
where

Ω i : Q × p → E × is defined by Ω i := U λ i 0 ,j 0 i /p n-1 • ω k i 0 ,j 0 i for all i ∈ {0, 1, • • • , n -1}. Moreover, FL i0,j0 n (ρ 0 ) = i0-1 i=j0+1 Ω -1 i (p) p (i 0 +j 0 )(i 0 -j 0 -1) 2 ∈ P 1 (F).
Proof. Notice that φ is diagonal on D := D Qp st (ρ 0 ) with respect to a framed basis e := (e n-1 , • • • , e 0 ) (which satisfies ge i = ω k i 0 ,j 0 i (g)e i for all i and for all g ∈ Gal(K/Q p )) since ω k i 0 ,j 0 i are all distinct. Hence, we have WD(ρ 0 ) = WD(ρ 0 ) F-ss . Similarly, since the descent data action on D commutes with the monodromy operator N , it is immediate that N = 0.

From the definition of WD(ρ 0 ) (c.f. [CDT99]), the action of W Qp on D can be described as follows: let α(g) ∈ Z be determined by ḡ = φ α(g) , where φ is the arithmetic Frobenius in G Fp and ḡ is the image under the surjection

W Qp Gal(K/Q p ). Then WD(ρ 0 )(g) • e i = λ i0,j0 i p n-1 -α(g) • ω k i 0 ,j 0 i (g) • e i for all i ∈ {0, 1, • • • , n -1}. (Recall that D Qp,n-1 st (ρ 0 ) = D Qp st (ρ 0 ⊗ ε -(n-1)
), so that the λ i 0 ,j 0 i p n-1 are the Frobenius eigenvalues of the Frobenius on D.) Write Ω i for the eigen-character with respect to e i .

Recall that we identify the geometric Frobenius with the uniformizers in Q × p (by our normalization of class field theory), so that Ω i (p) = p n-1 λ i 0 ,j 0 i which completes the proof by applying Theorem 3.3.44.

Local automorphic side

In this section, we establish several results concerning representation theory of GL n , that will be applied to the proof of our main results on mod p local-global compatibility, Theorem 3.5.44. The main results in this section are the non-vanishing result, Corollary 3.4.37, as well as the intertwining identity in characteristic 0, Theorem 3.4.71.

We start this section by fixing some notation. Let G := GL n/Zp and T be the maximal split torus consisting of diagonal matrices. We fix a Borel subgroup B ⊆ G consisting of upper-triangular matrices, and let U ⊆ B be the maximal unipotent subgroup. Let Φ + denote the set of positive roots with respect to (B, T ), and ∆ = {α k } 1≤k≤n-1 the subset of positive simple roots. Let X(T ) and X ∨ (T ) denote the abelian group of characters and cocharacters respectively. We often say a weight for an element in X(T ), and write X(T ) + for the set of dominant weights. The set Φ + induces a partial order on X(T ): for λ, µ ∈ X(T ) we say that λ ≤ µ if µ -λ ∈ α∈Φ + Z ≥0 α. We will also write λ < µ if λ ≤ µ and λ = µ.

We use the n-tuple of integers λ = (d 1 , d 2 , • • • , d n ) to denote the character associated to the weight

n k=1 d k k of T where for each 1 ≤ i ≤ n i is a weight of T defined by diag(x 1 , x 2 , • • • , x n ) i → x i .
We will often use the following weight

η := (n -1, n -2, • • • , 1, 0).
We let G, B, • • • be the base change to F p of G, B, • • • respectively. The Weyl group of G, denoted by W , has a standard lifting inside G as the group of permutation matrix, likewise with G. We denote the longest Weyl element by w 0 which is lifted to the antidiagonal permutation matrix in G or G. We use the notation s i for the simple reflection corresponding to α i = i -i+1 for 1 ≤ i ≤ n -1. We define the length (w) of w ∈ W to be its minimal length of decomposition into product of s i for 1 ≤ i ≤ n -1. Given A ∈ U (F p ), we use A α or equivalently A i,j to denote the (i, j)-entry of A, where α = ij is the positive root corresponding to the pair (i, j) with 1 ≤ i < j ≤ n.

Given a representation π of G(F p ), we use the notation π µ for the T (F p )-eigenspace with the eigencharacter µ. Given an algebraic representation V of G or G, we use the notation V λ for the weight space of V associated to the weight λ. For any representation V of G or G(F p ) with coefficient in F p , we define

V F := V ⊗ Fp F
to be the extension of coefficient of V from F p to F. Similarly, we write

V Fp for V ⊗ Fp F p .
It is easy to observe that we can identify the character group of T (F p ) with X(T )/(p -1)X(T ). The natural action of the Weyl group W on T and thus on T (F p ) induces an action of W on the character group X(T ) and X(T )/(p -1)X(T ). We carefully distinguish the notation between them. We use the notation wλ (resp. µ w ) for the action of W on X(T ) (resp. X(T )/(p -1)X(T )) satisfying

wλ(x) = λ(w -1 xw) for all x ∈ T and µ w (x) = µ(w -1 xw) for all x ∈ T (F p ).
As a result, without further comments, the notation wλ is a weight but µ w is just a character of T (F p ). There is another dot action of W on X(T ) defined by w • λ = w(λ + η) -η for all λ ∈ X(T ) and w ∈ W.

The affine Weyl group W of G is defined as the semi-direct product of W and X(T ) with respect to the natural action of W on X(T ). We denote the image of λ ∈ X(T ) in W by t λ . Then the dot action of W on X(T ) extends to the dot action of W on X(T ) through the following formula

w • λ = w • (λ + pµ) if w = wt µ . We use the notation λ ↑ µ for λ, µ ∈ X(T ) if λ ≤ µ and λ ∈ W • µ. We define a specific element of W by w h := w 0 t -η
following Section 4 of [START_REF] Le | Weight elimination in Serre type conjectures[END_REF]. We usually write K for GL n (Z p ) for brevity. We will also often use the following three open compact subgroups of GL n (Z p ): if we let red : GL n (Z p )

GL n (F p ) be the natural mod p reduction map, then

K(1) := Ker(red) ⊂ I(1) := red -1 (U (F p )) ⊂ I := red -1 (B(F p )) ⊂ K.
For each α ∈ Φ + , there exists a subgroup U α of G such that xu α (t)x -1 = u α (α(x)t) where x ∈ T and u α : G a → U α is an isomorphism sending 1 to 1 in the entry corresponding to α. For each α ∈ Φ + , we have the following equalities by [Jan03] II.1.19 (5) and (6):

u α (t) = m≥0 t m (X alg α,m ). (3.4.1)
where X alg α,m is an element in the algebra of distributions on G supported at the origin 1 ∈ G. The equation (3.4.1) is actually just the Taylor expansion with respect to t of the operation u α (t) at the origin 1. We use the same notation X alg α,m if G is replaced by G. We define the set of p-restricted weights as

X 1 (T ) := {λ ∈ X(T ) | 0 ≤ λ, α ∨ ≤ p -1 for all α ∈ ∆}
and the set of central weights as

X 0 (T ) := {λ ∈ X(T ) | λ, α ∨ = 0 for all α ∈ ∆}.
We also define the set of p-regular weights as

X reg 1 (T ) := {λ ∈ X(T ) | 1 ≤ λ, α ∨ ≤ p -2 for all α ∈ ∆},
and in particular we have X reg 1 (T ) X 1 (T ). We say that λ ∈ X(T ) lies in the lowest p-restricted

alcove if 0 < λ + η, α ∨ < p for all α ∈ Φ + . (3.4.2)
We define a subset W + of W as

W + := { w ∈ W | w • λ ∈ X(T ) + for each λ in the lowest p-restricted alcove}.
We define another subset W res of W as

W res := { w ∈ W | w • λ ∈ X 1 (T )
for each λ in the lowest p-restricted alcove}.

(3.4.3)

In particular, we have the inclusion

W res ⊆ W + .
For any weight λ ∈ X(T ), we let

H 0 (λ) := Ind G B w 0 λ alg /Fp
be the associated dual Weyl module. Note by [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF], Proposition II.2.6 that H 0 (λ) = 0 if and only if λ ∈ X(T ) + . Assume that λ ∈ X(T ) + , we write F (λ) := soc G (H 0 (λ)) for its irreducible socle as an algebraic representation (c.f. [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] part II, section 2). When λ is running through X 1 (T ), the F (λ) exhaust all the irreducible representations of G(F p ). On the other hand, two weights λ, λ ∈ X 1 (T ) satisfies

F (λ) ∼ = F (λ ) as G(F p )-representation if and only if λ -λ ∈ (p -1)X 0 (T ).
If λ ∈ X reg 1 (T ), then the structure of F (λ) as a G(F p )-representation depends only on the image of λ in X(T )/(p -1)X(T ), namely as a character of T (F p ). Conversely, given a character µ of T (F p ) which lies in the image of X reg 1 (T ) → X(T )/(p -1)X(T ), its inverse image in X reg 1 (T ) is uniquely determined up to a translation of (p -1)X 0 (T ). In this case, we say that µ is p-regular. Whenever it is necessary for us to lift an element in X(T )/(p -1)X(T ) back into X 1 (T ) (or maybe X reg 1 (T )), we will clarify the choice of the lift. Consider the standard Bruhat decomposition

G = w∈W BwB = w∈W U w wB = w∈W BwU w -1 .
where U w is the unique subgroup of U satisfying BwB = U w wB as schemes over Z p . The group U w has a unique form of α∈Φ + w U α for the subset Φ + w of Φ + defined by Φ + w = {α ∈ Φ + , w(α) ∈ -Φ + }. (If w = 1, we understand α∈Φ + w U α to be the trivial group 1.) We also have the following Bruhat decomposition:

G(F p ) = w∈W B(F p )wB(F p ) = w∈W U w (F p )wB(F p ) = w∈W B(F p )wU w -1 (F p ).
(3.4.4)

Given any integer x, recall that we use the notation

Jacobi sums in characteristic p

In this section we establish several fundamental properties of Jacobi sum operators on mod p principal series representations.

Definition 3.4.5. A weight λ ∈ X(T ) is called k-generic for k ∈ Z >0 if for each α ∈ Φ + there exists m α ∈ Z such that m α p + k < λ, α ∨ < (m α + 1)p -k.
In particular, the n-tuple of integers

(a n-1 , • • • , a 1 , a 0 ) is called k-generic in the lowest alcove if a i -a i-1 > k ∀ 1 ≤ i ≤ n -1 and a n-1 -a 0 < p -k.
Note that (a n-1 , • • • , a 0 )-η lies the lowest restricted alcove in the sense of (3.4.2) if (a n-1 , • • • , a 0 ) is k-generic in the lowest alcove for some k > 0. Note also that the existence of a n-tuple of integers satisfying k-generic in the lowest alcove implies p > n(k + 1) -1.

We use the notation π for a general principal series representation:

π := Ind G(Fp) B(Fp) µ π = {f : G(F p ) → F p | f (bg) = µ π (b)f (g) ∀(b, g) ∈ B(F p ) × G(F p )}
where µ π is a mod p character of T (F p ). The action of GL n (F p ) on π is given by (g • f )(g ) = f (g g).

We will assume throughout this article that µ π is p-regular. By definition we have

cosoc G(Fp) (π) = F (µ π ) and soc G(Fp) (π) = F (µ w0 π ).
By Bruhat decomposition we can deduce that dim Fp π U (Fp),µ w π = 1 for each w ∈ W . We denote by v π a non-zero fixed vector in π U (Fp),µπ . We also consider the natural lift π • of π defined as

π • := Ind G(Fp) B(Fp) µ π = {f : G(F p ) → Z p | f (bg) = µ π (b)f (g) ∀(b, g) ∈ B(F p ) × G(F p )} (3.4.6)
where µ π is the Teichmüller lift of µ π . Given w ∈ W with w = 1 and k

= (k α ) α∈Φ + w ∈ {0, 1, • • • , p -1} |Φ + w | , we define the Jacobi sum operators S k,w := A∈Uw(Fp)   α∈Φ + w A kα α   A • w ∈ F p [G(F p )].
(3.4.7)

These Jacobi sum operators play a main role on the local automorphic side as a crucial computation tool. These operators already appeared in [START_REF] Carter | Modular representations of finite groups of Lie type[END_REF] for example. For each α ∈ Φ + and each integer m satisfying 0 ≤ m ≤ p -2, we define the operator

X α,m := t∈Fp t p-1-m u α (t) ∈ F p [U (F p )] ⊆ F p [G(F p )]. (3.4.8)
The transition matrix between {u α (t

) | t ∈ F × p } and {X α,m | 0 ≤ m ≤ p -2} is a Vandermonde matrix t k t∈F × p ,1≤k≤p-1
which has a non-zero determinant. Hence, we also have a converse formula

u α (t) = - p-2 m=0
t m X α,m for all t ∈ F p .

(3.4.9)

By the equation (3.4.1), we note that we have the equality

X α,m = - k≥0 X alg α,m+(p-1)k . (3.4.10) Lemma 3.4.11. Fix w ∈ W and α 0 = (i 0 , j 0 ) ∈ Φ + w . Given a tuple of integers k = (k i,j ) ∈ {0, 1, • • • , p -1} |Φ + w | satisfying
k i0,j = 0 for all (i 0 , j) ∈ Φ + w with j ≥ j 0 + 1, (3.4.12)

we have

X α0,m • S k,w = (-1) m+1 c kα 0 ,m S k ,w if m ≤ k α0 0 if m > k α0
where k = (k α ) α∈Φw satisfies

k α = k α0 -m if α = α 0 ; k α otherwise.
Proof. We prove this lemma by direct computation.

X α,m • S k,w = t∈Fp t p-1-m   A∈Uw(Fp)   α∈Φ + w A kα α   u α0 (t)Aw   = t∈Fp t p-1-m   A∈Uw(Fp)   α∈Φ + w ,α =α0 A kα α   (A α0 -t) kα 0 Aw   = A∈Uw(Fp)   α∈Φ + w ,α =α0 A kα α     t∈Fp t p-1-m (A α0 -t) kα 0   Aw (3.4.13)
where the second equality follows from the change of variable A ↔ u α0 (t)A and the assumption (3.4.12). Notice that

t∈Fp t p-1-m (A α0 -t) kα 0 = t∈Fp t p-1-m   kα 0 j=0 (-1) j c kα 0 ,j A kα 0 -j kα 0 t j   = kα 0 j=0 (-1) j c kα 0 ,j A kα 0 -j kα 0   t∈Fp t p-1-m+j   ,
which can be easily seen to be

(-1) m+1 c kα 0 ,m A kα 0 -m kα 0 if m ≤ k α0 0 if m > k α0 . ( 3
= (i 0 , j 0 ) ∈ Φ + w . Given a tuple of integers k = (k i,j ) ∈ {0, 1, • • • , p -1} |Φ + w | satisfying k i0,j = 0 for all (i 0 , j) ∈ Φ + w with j ≥ j 0 , we have u α0 (t) • S k,w = S k,w .
Proof. By Lemma 3.4.11 we deduce that

X α0,m • S k,w = -S k,w if m = 0 0 if 1 ≤ m ≤ p -2
Therefore we conclude this lemma from (3.4.9).

By the definition of principal series representations, we have the decomposition π = ⊕ w∈W π w (3.4.17)

where π w ⊂ π| B(Fp) consists of the functions supported on a non-empty subset of the Bruhat cell

B(F p )w -1 B(F p ) = B(F p )w -1 U w (F p ).
Proposition 3.4.18. Fix a non-zero vector v π ∈ π U (Fp),µπ . For each w ∈ W with w = 1, the set

S k,w v π | k = (k α ) α∈Φ + w ∈ {0, 1, • • • , p -1} |Φ + w |
forms a T (F p )-eigenbasis of π w .

Proof. We have a decomposition π w = ⊕ A∈Uw(Fp) π w,A where π w,A is the subspace of π w consisting of functions supported on B(F p )w -1 A -1 . It is easy to observe by the definition of parabolic induction that dim Fp π w,A = 1 and π w,A is generated by Awv π . We claim that the set of Jacobi sums with the Weyl element w, after being applied to v π , differs from the set {Awv π , A ∈ U w (F p )} by an invertible matrix. More precisely, for a fixed w ∈ W , the set of vectors

{S k,w v π | k = (k α ) α∈Φ + w ∈ {0, 1, • • • , p -1} |Φ + w | } can be linearly represented by the set of vectors {Awv π , A ∈ U w (F p )} through the matrix m k,A where k = (k α ) α∈Φ + w ∈ {0, 1, • • • , p -1} |Φ + w | , A ∈ U w (F p )
and m k,A := α∈Φ + w A kα α . Note that this matrix is the |Φ + w |-times tensor of the Vandermonde matrix

λ k λ∈Fp,0≤k≤p-1 , 80CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
and therefore has a non-zero determinant. As a result, the matrix m k,A is invertible and {S k,w v π | 0 ≤ k α ≤ p -1 ∀α ∈ Φ + w } forms a basis of π w . The fact that this is a T (F p )-eigenbasis is immediate by the following calculation: if we let x

= diag(x 1 , x 2 , • • • , x n ) x • S k,w v π = x •   A∈Uw(Fp)   α∈Φ + w A kα α   A w   v π =   A∈Uw(Fp)   (i,j)∈Φ + w A ki,j i,j   xAx -1 w   w -1 xw v π =   B=xAx -1 ∈Uw(Fp)   (i,j)∈Φ + w (B i,j x j x -1 i ) ki,j   B w   w -1 xw v π = µ π (w -1 xw)   (i,j)∈Φ + w (x j x -1 i ) ki,j     A∈Uw(Fp) α∈Φ + w A kα α A w   v π = (µ w π λ)(x)S k,w v π ,
where λ(x) = 1≤i<j≤n (x j x -1 i ) ki,j and B i,j = A i,j x i x -1

j for 1 ≤ i < j ≤ n.
We can further describe the action of T (F p ) on S k,w v π . By y for y ∈ R we mean the floor function of y, i.e., the biggest integer less than or equal to y.

Lemma 3.4.19. Let µ π = (d 1 , d 2 , • • • , d n-1 , d n ). If we write ( 1 , 2 • • • , n-1 , n ) for the T (F p )- eigencharacter of S k,w v π , then we have r ≡ d w -1 (r) + 1≤i<r k i,r - r<j≤n k r,j (mod p -1)
for all 1 ≤ r ≤ n, where k i,j = k α if α ∈ Φ + w and (i, j) corresponds to α, and k i,j = 0 otherwise. In particular,

(i) if k α = 0 for any α ∈ Φ + w \ ∆, then for all 1 ≤ r ≤ n r ≡ d w -1 (r) + (1 -1/r )k r-1,r -(1 -1/(n + 1 -r) )k r,r+1 (mod p -1); (ii) if w = w 0 and k i,j = 0 for any 2 ≤ i < j ≤ n, then r ≡ d n - n j=2 k 1,j (mod p -1) if r = 1; d n+1-r + k 1,r (mod p -1) if 2 ≤ r ≤ n.
Proof. The first part of the Lemma is a direct calculation as shown at the end of the proof of Proposition 3.4.18. The second part follows trivially from the first part.

Given any w ∈ W , we write S 0,w for S k,w with k α = 0 for all α ∈ Φ + w .

Lemma 3.4.20. F p [S 0,w v π ] = π U (Fp),µ w π .

Proof. Pick an arbitrary positive root α. If α ∈ Φ + w , then we have (since

u α (t) ∈ U w (F p )) u α (t)   A∈Uw(Fp) A   =   A∈Uw(Fp) A  
and therefore u α (t)S 0,w v π = S 0,w v π for any t ∈ F p . On the other hand, if α / ∈ Φ + w , then

u α (t)   A∈Uw(Fp) A   =   A∈Uw(Fp) A   u α (t)
and u α (t)wv π = wu α (t)v π = wv π where u α (t) ∈ α / ∈Φ + w U α (F p ) and u α (t) ∈ U (F p ) are elements depending on x, w and α. Hence, u α (t)S 0,w v π = S 0,w v π for any t ∈ F p and any α ∈ Φ + . So we conclude that S 0,w v π is U (F p )-invariant as {u α (t)} α∈Φ + ,t∈Fp generate U (F p ).

Finally, we check that x • S 0,w v π = µ w π (x)S 0,w v π for x ∈ T (F p ). But this is immediate from the following two easy computations:

x •   A∈Uw(Fp) A   =   A∈Uw(Fp) A   • x ∈ F p [G(F p )]
and

xwv π = w w -1 xw v π = wµ π (w -1 xw)v π = µ w π (x)wv π .
This completes the proof.

Note that Proposition 3.4.18, Lemma 3.4.19, and Lemma 3.4.20 are very elementary and have essentially appeared in [START_REF] Carter | Modular representations of finite groups of Lie type[END_REF]. In this article, we formulate them and give quick proofs of them for the convenience. Definition 3.4.21. Given α, α ∈ Φ + , we say that α is strongly smaller than α with the notation

α ≺ α if there exist 1 ≤ i ≤ j ≤ k ≤ n -1 such that α = j r=i α r and α = k r=i α r .
We call a subset Φ of Φ + good if it satisfies the following:

(i) if α, α ∈ Φ and α + α ∈ Φ + , then α + α ∈ Φ ; (ii) if α ∈ Φ and α ≺ α , then α ∈ Φ .
We associate a subgroup of U to Φ by

U Φ := U α | α ∈ Φ (3.4.22)
and denote its reduction mod p by U Φ . We define U 1 to be the subgroup scheme of U generated by U αr for 2 ≤ r ≤ n -1, and denote its reduction mod p by U 1 .

Example 3.4.23. The following are two examples of good subsets of Φ + , that will be important for us:

j r=i α r | 1 ≤ i < j ≤ n -1 and j r=i α r | 2 ≤ i ≤ j ≤ n -1 .
The subgroups of U associated with the two good subsets via (3.4.22) are [U, U ] and U 1 respectively.

82CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
We recall that we have defined π w π in (3.4.17) for each w ∈ W .

Proposition 3.4.24. Let Φ ⊆ Φ + be good. Pick an element w ∈ W with w = 1. The following set of vectors

S k,w v π | k = (k α ) α∈Φ + w ∈ {0, 1, • • • , p -1} |Φ + w | with k α = 0 ∀α ∈ Φ ∩ Φ + w (3.4.25)
forms a basis of the subspace π

U Φ (Fp) w of π w .
Proof. By Proposition 3.4.18, the Jacobi sums with the Weyl element w, after being applied to v π , form a T (F p )-eigenbasis of π w , and so we can firstly write any U Φ (F p )-invariant vector v in π w as a unique linear combination of Jacobi sums with the Weyl element w,

namely v = k∈{0,••• ,p-1} |Φ + w | C k,w S k,w v π for some C k,w ∈ F p .
Assume that C k,w = 0 for certain tuple of integers k = (k α ) α∈Φ + w such that k α > 0 for some α ∈ Φ ∩ Φ + w . We choose α 0 such that it is maximal with respect to the partial order ≺ on Φ + for the property C k,w = 0, k α0 > 0, and

α 0 ∈ Φ ∩ Φ + w . (3.4.26)
We may write v as follows:

v = k∈{0,••• ,p-1} |Φ + w | kα 0 =0 C k,w S k,w v π + k∈{0,••• ,p-1} |Φ + w | kα 0 >0 C k,w S k,w v π . (3.4.27)
By the maximal assumption on α 0 we know that if C k,w = 0 and α 0 ≺ α, then k α = 0. As a result, we deduce from Lemma 3.4.16 that

u α0 (t) k∈{0,••• ,p-1} |Φ + w | kα 0 =0 C k,w S k,w v π = k∈{0,••• ,p-1} |Φ + w | kα 0 =0 C k,w S k,w v π ( 3
) :=    k = (k α ) α∈Φ + w ∈ {0, • • • , p -1} |Φ + w | • k α = 0 if α ∈ Φ α0,+ w \ {α 0 }; • k α > 0 if α = α 0 ; • k α = α if α ∈ Φ α0,- w    . Now we can define a polynomial f ( ,α0) (x) = k∈Λ( ,α0) C k,w x kα 0 ∈ F p [x]
for each tuple of integers . By definition, we have

k∈{0,••• ,p-1} |Φ + w | kα 0 >0 C k,w S k,w v π = ∈{0,••• ,p-1} |Φ α 0 ,- w |   A∈Uw(Fp)   α∈Φ α 0 ,- w A α α   f ( ,α0) (A α0 )A   wv π .
By the assumption on v we know that u α0 (t)v = v for all t ∈ F p . Using (3.4.28) and (3.4.27) we have

u α0 (t) k∈{0,••• ,p-1} |Φ + w | kα 0 >0 C k,w S k,w v π = k∈{0,••• ,p-1} |Φ + w | kα 0 >0 C k,w S k,w v π and so ∈{0,••• ,p-1} |Φ α 0 ,- w |   A∈Uw(Fp)   α∈Φ α 0 ,- w A α α   f ( ,α0) (A α0 )A   wv π = u α0 (t) ∈{0,••• ,p-1} |Φ α 0 ,- w |   A∈Uw(Fp)   α∈Φ α 0 ,- w A α α   f ( ,α0) (A α0 )A   wv π = ∈{0,••• ,p-1} |Φ α 0 ,- w |   A∈Uw(Fp)   α∈Φ α 0 ,- w A α α   f ( ,α0) (A α0 -t)A   wv π
where the last equality follows from a change of variable A ↔ u α0 (t)A. By the linear independence of Jacobi sums from Proposition 3.4.18, we deduce an equality

  A∈Uw(Fp)   α∈Φ α 0 ,- w A α α   f ( ,α0) (A α0 )A   wv π =   A∈Uw(Fp)   α∈Φ α 0 ,- w A α α   f ( ,α0) (A α0 -t)A   wv π
for each fixed tuple . Therefore, again by the linear independence of Jacobi sum operators in Proposition 3.4.18 we deduce that

f ( ,α0) (A α0 -t) = f ( ,α0) (A α0 )
for all t ∈ F p and each ( , α 0 ). We know that if f ∈ F p [x] satisfies degf ≤ p -1, f (0) = 0 and f (x -t) = f (x) for all t ∈ F p then f = 0. Thus we deduce that f ( ,α0) = 0 for each tuple of integers , which is a contradiction to (3.4.26) and so we have k α = 0 for any α ∈ Φ for each tuple of integers k such that C k,w = 0. As a result, we have shown that each vector in π U Φ (Fp) w can be written as certain linear combination of vectors in (3.4.25). On the other hand, by Proposition 3.4.18 we know that vectors in (3.4.25) are linear independent, and therefore they actually form a basis of π

U Φ (Fp) w . Corollary 3.4.29. Let µ π = (d 1 , • • • , d n ) and fix a non-zero vector v π ∈ π U (Fp),µπ . Given a weight µ = ( 1 , • • • , n ) ∈ X 1 (T ) the space π [U (Fp),U (Fp)],µ w0
has a basis whose elements are of the form

S k,w0 v π where k = (k α ) satisfies r ≡ d n+1-r + (1 -1/r )k r-1,r -(1 -1/(n + 1 -r) )k r,r+1 mod (p -1) for all 1 ≤ r ≤ n and k α = 0 if α ∈ Φ + \ ∆. 84CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Proof. By a special case of Proposition 3.4.25 when Φ = { j r=i α r | 1 ≤ i < j ≤ n -1}, we know that {S k,w0 v π | k α = 0 if α ∈ Φ + \ ∆} forms a basis of π [U (Fp),U (Fp)] w0
. On the other hand, we know from Proposition 3.4.18 that the above basis is actually an T (F p )-eigenbasis. Therefore the vectors in this basis with a fixed eigencharacter µ form a basis of the eigensubspace π [U (Fp),U (Fp)],µ w0 . Finally, using (i) of the second part of Lemma 3.4.19 we conclude this lemma.

Corollary 3.4.30. Let µ π = (d 1 , d 2 , • • • , d n ) and fix a non-zero vector v π ∈ π U (Fp),µπ . Given a weight µ = ( 1 , • • • , n ) ∈ X 1 (T ), the space π U1(Fp),µ w0
has a basis whose elements are of the form

S k,w0 v π where k = (k i,j ) i,j satisfies k 1,j ≡ j -d n+1-j mod (p -1)
for 2 ≤ j ≤ n and k i,j = 0 for all 2 ≤ i < j ≤ n.

Proof. By a special case of Proposition 3.4.25 when Φ = {

j r=i α r | 2 ≤ i ≤ j ≤ n -1}, we know that {S k,w0 v π | k i,j = 0 if 2 ≤ i < j ≤ n} forms a basis of π U1(Fp) w0
. On the other hand, we know from Proposition 3.4.18 that the above basis is actually an T (F p )-eigenbasis. Therefore the vectors in this basis with a fixed eigencharacter µ form a basis of the eigensubspace π U1(Fp),µ w0

. Finally, using (ii) of the second part of Lemma 3.4.19 we conclude this lemma.

Main results in characteristic p

In this section, we state our main results on certain Jacobi sum operators in characteristic p. From now on we fix an n-tuple of integers (a n-1 , • • • , a 0 ) which is assumed to be n-generic in the lowest alcove (c.f. Definition 3.4.5).

We let

       µ * := (a n-1 -n + 2, a n-2 , a n-3 , • • • , a 2 , a 1 , a 0 + n -2); µ 1 := (a 1 , a 2 , • • • , a n-3 , a n-2 , a n-1 , a 0 ); µ 1 := (a n-1 , a 0 , a 1 , a 2 , • • • , a n-3 , a n-2 ); µ 0 := (a n-1 , a 1 , a 2 • • • , a n-3 , a n-2 , a 0 ); (3.4.31) and π 0 := Ind G(Fp) B(Fp) µ 0 ; π • 0 := Ind G(Fp) B(Fp) µ 0 ; (3.4.32)
where µ 0 is the Teichmüller lift of µ 0 . We let k 1 = (k 1 i,j ), k 1, = (k 1, i,j ) and k 0 = (k 0 i,j ), where

   k 1 i,i+1 = [a 0 -a n-i ] 1 + n -2; k 1, i,i+1 = [a n-i-1 -a n-1 ] 1 + n -2; k 0 i,i+1 = [a 0 -a n-1 ] 1 + n -2;
(3.4.33)

for 1 ≤ i ≤ n -1 and k 1 i,j = k 1, i,j = k 0 i,j = 0 otherwise, and define two most important Jacobi sum operators S n and S n to be

S n := S k 1 ,w0
and S n := S k 1, ,w0 .

(3.4.34)

We also let V 1 , V 1 , and V 0 denote the sub-representation of π 0 generated by

S n π U (Fp),µ1 0 , S n π U (Fp),µ 1 0
, and S k 0 ,w0 π U (Fp),µ0 0

respectively.

The following theorem, which we usually call the non-vanishing theorem, is a technical heart on the local automorphic side. The proofs of this non-vanishing theorem as well as the next theorem, which we usually call the multiplicity one theorem, will occupy the following sections.

Theorem 3.4.35. Assume that (a n-1 , • • • , a 0 ) is n-generic in the lowest alcove.

Then we have

V 1 = V 1 = V 0 and F (µ * ) ∈ JH(V 0 ).
Proof. This is an immediate consequence of Corollary 3.4.67 and Theorem 3.4.137.

We also have the following multiplicity one result.

Theorem 3.4.36. Assume that (a n-1 , • • • , a 0 ) is 2n-generic in the lowest alcove.

Then F (µ * ) has multiplicity one in π 0 .

Proof. This is a special case of Corollary 3.4.47: replace µ 0,n-1 π with µ * .

Corollary 3.4.37. Assume that (a n-1 , • • • , a 0 ) is 2n-generic in the lowest alcove and that τ is an

O E -lattice in π • 0 ⊗ O E E such that soc G(Fp) (τ ⊗ O E F) = F (µ * ).
Then we have

S n (τ ⊗ O E F) U (Fp),µ1 = 0 and S n (τ ⊗ O E F) U (Fp),µ 1 = 0.
Proof. Such a τ is unique up to homothety by Theorem 3.4.36. By multiplying a suitable power of E , we may assume that

π • 0 τ and π • 0 τ,
and thus we have a non-zero morphism

π 0 → τ ⊗ O E F
whose image is the unique quotient of π 0 with socle F (µ * ). We now finish the proof by applying Theorem 3.4.35.

Summary of results on Deligne-Lusztig representations

In this section, we recall some standard facts on Deligne-Lusztig representations and fix the notation that will be used throughout this paper. We closely follow [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF]. Throughout this article we will only focus the group G(F p ) = GL n (F p ), which is the fixed point set of the standard (p-power) Frobenius F inside GL n (F p ). We will identify a variety over F p with the set of its F p -rational points for simplicity. Then our fixed maximal torus T is F -stable and split.

To each pair (T, θ) consisting of an F -stable maximal torus T and a homomorphism θ :

T F → Q × p , Deligne-Lusztig [DL76] associate a virtual representation R θ
T of GL n (F p ). (We restrict ourself to GL n (F p ) although the result in [START_REF] Deligne | Representations of reductive groups over finite fields[END_REF] is much more general.) On the other hand, given a pair (w, µ) ∈ W × X(T ), one can construct a pair (T w , θ w,µ ) by the method in the third paragraph of [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Section 4.1. Then we denote by R w (µ) the representation corresponding to R θw,µ Tw after multiplying a sign. This is the so-called Jantzen parametrization in [START_REF] Jantzen | Zur Reduktion modulo p der Charaktere von Deligne und Lusztig[END_REF] 3.1.

The representations R θ T (resp. R w (µ)) can be isomorphic for different pairs (T, θ) (resp. (w, µ)), and the explicit relation between is summarized in [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Lemma 4.2. As each p-regular character µ ∈ X(T )/(p-1)X(T ) of T (F p ) can be lift to an element in X reg 1 (T ) which is unique up to (p-1)X 0 (T ), the representation R w (µ) is well defined for each w ∈ W and such a µ.

We recall the notation Θ(θ) for a cuspidal representation for GL n (F p ) from [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Section 2.1 where θ is a primitive character of F × p n defined in [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Section 4.2. We refer further discussion about the basic properties and references of Θ(θ) to [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Section 2.1. The relation between the notation R w (µ) and the notation Θ(θ) is summarized in [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Lemma 4.7. In this paper, we will use the notation Θ m (θ m ) for a cuspidal representation for GL m (F p ) where θ m is a primitive character of F × p m . We emphasize that, as a special case of [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Lemma 4.7, we have the natural isomorphism

R 1 (µ) ∼ = Ind G(Fp) B(Fp) µ
for a p-regular character µ of T (F p ), where µ is the Teichmüler lift of µ.

Proof of Theorem 3.4.36

The main target of this section is to prove Theorem 3.4.36. In fact, we prove Corollary 3.4.47 which is a generalization of Theorem 3.4.36. We recall some notation from [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF]. We use the notation G r for the r-th Frobenius kernel defined in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] Chapter I.9 as kernel of r-th iteration of Frobenius morphism on the group scheme G over F p . We will consider the subgroup scheme G r T , G r B, G r B -of G in the following. Note that our B (resp. B -) is denoted by B + (resp. B) in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] Chapter II. 9. We define

Z r (λ) := ind GrB - B - λ; Z r (λ) := coind GrB B λ
where ind and coind are defined in I.3.3 and I.8.20 of [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] respectively. By [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] Proposition II.9.6 we know that there exists a simple G r T -module L r (λ) satisfying

soc Gr Z r (λ) ∼ = L r (λ) ∼ = cosoc Gr Z r (λ) .
The properties of Z r (λ) and Z r (λ) are systematically summarized in [Jan03] II.9, and therefore we will frequently refer to results over there.

From now on we assume r = 1 in this section. Now we recall several well-known results from [START_REF] Jantzen | Zur Reduktion modulo p der Charaktere von Deligne und Lusztig[END_REF], [START_REF] Jantzen | Filtrierungen Der Darstellungen in Der Hauptserie Endlicher Chevalley-Gruppen[END_REF] and [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF]. We recall the definition of W res from (3.4.3).

Proof. The condition (3.4.46) ensures that for any fixed ν ∈ X(T ), µ+pν is maximal for µ+pν ↑ λ+pν and µ + pν = λ + pν. Notice that we have

[ Z 1 (λ) : L 1 (µ)] = [ Z 1 (λ) : L 1 (µ)]
by II 9.2(3) in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF], as the character of a G r T -module determine its Jordan-Hölder factors with multiplicities (or equivalently, determine the semisimplification of the G r T -module).

By II 9.2(5) and II 9.6(6) in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] we have

[ Z 1 (λ) : L 1 (µ)] = [ Z 1 (λ) ⊗ pν : L 1 (µ) ⊗ pν] = [ Z 1 (λ + pν) : L 1 (µ + pν)],
and thus we may assume that λ, α ∨ ≥ n -2 for all α ∈ Φ + by choosing appropriate ν (which exists by our assumption) and replacing λ by λ + pν and µ by µ + pν. Then by Lemma 3.4.43 we know that

µ 1 + η, β ∨ ≥ 0 for any µ = pµ 1 + µ 0 such that L 1 (µ ) is a factor of Z 1 (λ).
Thus by Proposition 3.4.41, Proposition 3.4.40 and Remark 3.4.42 we know that

[ Z 1 (λ) : L 1 (µ)] = [H 0 (λ) : F (µ 0 ) ⊗ H 0 (µ 1 ) [1] ] = [H 0 (λ) : F (µ)] = 1
which finishes the proof.

We pick an arbitrary principal series π and write

µ π = (d 1 , • • • , d n )
For each pair of integers (i 1 , j 1 ) satisfying 0 ≤ i 1 < i 1 + 1 < j 1 ≤ n -1, we define

µ i1,j1 π := (d i1,j1 1 , • • • , d i1,j1 n )
where

d i1,j1 k =    d k if k = n -j 1 and k = n -i 1 ; d n-i1 + j 1 -i 1 -1 if k = n -i 1 ; d n-j1 -j 1 + i 1 + 1 if k = n -j 1 .
Corollary 3.4.47. Assume that µ π is 2n-generic in the lowest alcove (c.f. Definition 3.4.5). Then F (µ i1,j1 π ) has multiplicity one in π, or equivalently in Ind G(Fp) B(Fp) µ w π for any w ∈ W .

Proof. We notice at first that each Ind G(Fp) B(Fp) µ w π has the same Jordan-Hölder factor as π with the same multiplicity as each of them is a mod p reduction of certain lattice of the same characteristic zero representation of G(F p ). We only need to apply Theorem 3.4.45 and Theorem 3.4.39 to these explicit examples. We will follow the notation of Theorem 3.4.39. We fix w = 1 in Theorem 3.4.39 and take

µ + η := µ π = µ i1,j1 π + (j 1 -i 1 -1)   n-1-i1 r=n-j1 α r   .
We are considering the multiplicity of F (µ i1,j1 π ) in π = R 1 (µ + η) and therefore we take w := 1 ∈ W res and

ν := η -(j 1 -i 1 -1)   n-1-i1 r=n-j1 α r   .
By II. 9.2(4) and II.9.16 (4) in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] we know that

[ Z 1 (µ -pν + pη) : L 1 (µ)] = [ Z 1 ((n -j 1 , n -i 1 ) • (µ -pν) + pη) : L 1 (µ)].
(3.4.48)

We observe that

(n -j 1 , n -i 1 ) • (µ -pν) + pη = (n -j 1 , n -i 1 ) • µ + p   η -(n -j 1 , n -i 1 )η -(j 1 -i 1 -1)   n-1-i1 r=n-j1 α r     = (n -j 1 , n -i 1 ) • µ + p   n-1-i1 r=n-j1 α r   .
Therefore we have

p < (n -j 1 , n -i 1 ) • (µ -pν) + pη, n-1-i1 r=n-j1 α r < 2p and that µ = s n-1-i 1 r=n-j 1 αr,p • ((n -j 1 , n -i 1 ) • (µ -pν) + pη).
Moreover, it is easy to see that

(n -j 1 , n -i 1 ) • (µ -pν) + pη) + pη = (n -j 1 , n -i 1 ) • µ + p   n-1-i1 r=n-j1 α r   + pη satisfies (3.4.44).
Hence, replacing the λ and µ in Theorem 3.4.45 by (n -j 1 , n -i 1 ) • (µ -pν) + pη and µ respectively, we conclude that

[ Z 1 ((n -j 1 , n -i 1 ) • (µ -pν) + pη) : L 1 (µ)] = 1
which finishes the proof by Theorem 3.4.39 and (3.4.48).

Jacobi sums in characteristic 0

In this section, we establish an intertwining identity for lifts of Jacobi sums in characteristic 0 in Theorem 3.4.71, which is one of the main ingredients of the proof of Theorem 3.5.44. All of our calculations here are in the setting of G(Q p ) = GL n (Q p ). We first fix some notation.

Let A ∈ G(F p ). By A we mean the matrix in G(Q p ) whose entries are the classical Teichmüller lifts of the entries of A. The map A → A is obviously not a group homomorphism but only a map between sets. On the other hand, we use the notation µ for the Teichmüller lift of a character µ of T (F p ).

We denote the standard lifts of simple reflections in G(Q p ) by

s i =     Id i-1 1 1 Id n-i-1    
for 1 ≤ i ≤ n -1. We also use the following notation

t i = pId i Id n-i 90CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE for 1 ≤ i ≤ n. Let Ξ n := w * t 1 , (3.4.49)
where w * := s n-1 • ... • s 1 . We recall the Iwahori subgroup I and the pro-p Iwahori subgroup I(1) from the beginning of Section 3.4. Note that the operator Ξ n and the group I actually generates the normalizer of I inside G(Q p ). One easily sees that Ξ n is nothing else than the following matrix:

Ξ n =            0 1 0 • • • 0 0 0 0 0 1 • • • 0 0 0 0 0 0 • • • 0 0 0 . . . . . . . . . . . . . . . . . . . . . 0 0 0 • • • 0 1 0 0 0 0 • • • 0 0 1 p 0 0 • • • 0 0 0            ∈ G(Q p ).
For each 1 ≤ i ≤ n -1, we consider the maximal parabolic subgroup P - i of G containing lowertriangular Borel subgroup B -such that its Levi subgroup can be chosen to be GL i × GL n-i which embeds into G in the standard way. We denote the unipotent radical of P - i by N - i . Then we introduce

U i n = A∈N - i (Fp) t -1 i A for each 1 ≤ i ≤ n -1. (3.4.50) Note that each A ∈ N - i has the form Id i 0 (n-i)×i * i×(n-i)
Id n-i .

for each 1 ≤ i ≤ n -1.

For each w ∈ W and each tuple k = (k α ) α∈Φ + w ∈ {0, . . . , p -1} |Φ + w | , we consider the following Jacobi sum

S k,w :=   A∈Uw(Fp)   α∈Φ + w A α kα   A   w ∈ Z p [G(Z p )].
In particular, we consider

S w :=   A∈Uw(Fp) A   w ∈ Z p [G(Z p )]
which is a characteristic zero lift of S 0,w . Recall the notation π • from (3.4.6).

Lemma 3.4.51. Assume that µ π is n-generic (Definition 3.4.5). We have the equality The equality (3.4.52) follows directly from the definition of the Jacobi sum operators. The equality (3.4.53) follows from a simple Bruhat decomposition. In fact, we have for each t = 0

S w • S w = p
s r u αr (t)s r = u αr (t -1 )s r diag(1, • • • , 1, t, -t -1 , 1, • • • , 1)u αr (t -1 )
where the diagonal matrix has t at (r, r)-entry and -t -1 at (r + 1, r + 1)-entry. Therefore for each v ∈ ( π • ) I(1) there exists an integer n ≤ ≤ p -n such that

diag(1, • • • , 1, t, -t -1 , 1, • • • , 1) v = ± t v
and thus

S sr • S sr v =   t ∈Fp u αr (t )     v ± t∈F p t u αr (t -1 ) s r v   = p v ±   t ∈Fp,t∈Fp t u αr (t + t -1 ) s r   v = p v.
This finishes the proof.

Lemma 3.4.54. We have the equality

(Ξ n ) k • U k n = S (w * ) k
Proof. This is immediate by definition.

We quickly recall some standard facts about Jacobi sums and Gauss sums. We fix a primitive p-th root of unity ξ ∈ E and set := ξ -1. for each integers a with 0 ≤ a ≤ p -1. For example, we have G(p -1) = -1. It is known by section 1.1, GS3 of [Lang] that if a + b ≡ 0 mod (p -1), we have

J(a, b) = G(a)G(b) G(a + b) . (3.4.56)
It is also obvious from the definition that if a, b, a + b ≡ 0 mod (p -1) then

J(b, a) = J(a, b) = (-1) b J(b, [-a -b] 1 ) = (-1) a J(a, [-a -b] 1 ). ( 3
k α = k α if α ∈ Φ + w ; 0 if α = wα r
in the first case and

k α = [k wαr -d r + d r+1 ] 1 if α = wα r ; k α if α ∈ Φ + w and α = wα r
in the second case.

Proposition 3.4.60. Assume that

µ π = (d 1 , d 2 , • • • , d n ) is n-generic and that k α = 0 for all α ∈ Φ + w with wα r < α. Assume further that if (ws r ) < (w) then k wαr ∈ {0, p -1, [d r -d r+1 ] 1 }. Then for each 1 ≤ r ≤ n -1 we have S k,w • S sr = S k ,wsr if (ws r ) > (w) (-1) dr+1 J(k wαr , [d r+1 -d r ] 1 ) S k ,w if (ws r ) < (w) on ( π • ) I(1), µπ .
Proof. By definition we have

S k,w • S sr = A∈U (Fp),t∈Fp   α∈Φ + w A α kα   A w u αr (t) s r .
We divide it into two cases: (i) (ws r ) > (w);

(ii) (ws r ) < (w).

In case (i), we have the Bruhat decomposition Awu αr (t)s r = Au wαr (t)ws r and thus S k,w • S sr = S k ,wsr .

In case (ii), we have the Bruhat decompositions: if t = 0

Awu αr (0)s r = A(ws r ) = A ws r u αr (A wαr )

where A is the unipotent matrix that has the same entries as A except a zero at wα r -entry;

if t = 0 Awu αr (t)s r = Au wαr (t -1 )wdiag(1, • • • , t, -t -1 , • • • , 1)u αr (t -1 ).
We fix a vector v π ∈ ( π • ) I(1), µπ whose mod p reduction is non-zero. Therefore, we have

S k,w • S sr v π = (-1) dr+1 A∈Uw(Fp),t∈F × p   α∈Φ + w A α kα   t dr-dr+1 A u wαr (t -1 )w v π + A∈Uw(Fp)   α∈Φ + w A α kα   A ws r v π .
The summation A∈Uw(Fp) α∈Φ + w A α kα Aws r v π can be rewritten as

A ∈Uws r (Fp)   α∈Φ + wsr A α kα     Awα r ∈Fp A wαr kwα r   A ws r v π
which is 0 as we assume 0 < k wαr < p -1. Hence, we have

S k,w0 • S sr v π = (-1) dr+1 A∈Uw(Fp),t∈F × p   α∈Φ + w A α kα   t dr-dr+1 Au αwα r (t -1 ) w v π .
On the other hand, after setting A = Au wαr (t -1 ) we have

A∈Uw(Fp),t∈F × p   α∈Φ + w A α kα   t dr-dr+1 Au wαr (t -1 ) w v π = A ∈Uw(Fp),t∈F × p   α∈Φ + wsr A α kα   (A wαr -t -1 ) kwα r t dr-dr+1 A w v π (3.4.61)
since k α = 0 for all wα r < α.

One can easily check that if A wαr = 0 then t∈F × p (A wαr -t -1 ) kwα r t dr-dr+1 = (-1) kwα r t∈Fp t dr-dr+1-kwα r = 0, and if A wαr = 0 then

t∈F × p (A wαr -t -1 ) kwα r t dr-dr+1 = A wαr kwα r -dr+dr+1   t∈Fp (1 -(A wαr t) -1 ) kwα r (A wαr t) -1 ) dr+1-dr   = J(k wαr , [d r+1 -d r ] 1 ) A wαr [kwα r -dr+1+dr]1
Combining these computations with (3.4.61) finishes the proof.

Remark 3.4.62. Proposition 3.4.60 is the technical heart of this section. It roughly says that [U (F p ), U (F p )]-invariant vectors behave well under intertwining of principal series, which is essentially why the identities in Theorem 3.4.71 and Proposition 3.5.25 exist. On the other hand, it is crucial that the vector v π is invariant under u αr (t) for t ∈ F p .

Recall the definition of µ 1 and µ 1 from (3.4.31). We recursively define sequences of elements in the Weyl group W by

w 1 = 1, w m = s n-m w m-1 ; w 1 = 1, w m = s m w m-1
for all 2 ≤ m ≤ n -1, where s m are the reflection of the simple roots α m . We define sequences of characters of T (F p )

µ m = µ wm 1 and µ m = (µ 1 ) w m 94CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
for all 1 ≤ m ≤ n -1. We also define several families of Jacobi sums:

S k m ,w0 and S k m, ,w0
for all integers m with 1 ≤ m ≤ n -1, where k m = (k m i,j ) satisfies

k m i,j =    n -2 + [a 0 -a n-1 ] 1 if 1 ≤ i = j -1 ≤ m; n -2 + [a 0 -a n-i ] 1 if m + 1 ≤ i = j -1 ≤ n -1; 0 otherwise and k m, = (k m, i,j ) satisfies k m, i,j =    n -2 + [a n-i-1 -a n-1 ] 1 if 1 ≤ i = j -1 ≤ n -m -1; n -2 + [a 0 -a n-1 ] 1 if n -m ≤ i = j -1 ≤ n -1; 0 otherwise.
We keep the notation in (3.4.32) and recall that k 0 is defined in (3.4.33). We also define κ

(1)

n := (-1) n-2 m=1 am n-2 m=1 J(n -2 + [a 0 -a n-m-1 ] 1 , [a n-m-1 -a n-1 ] 1 ), κ (2)
n := (-1) (n-2)a0 n-2 m=1 J(n -2 + [a m -a n-1 ] 1 , [a 0 -a m ] 1 ). (3.4.63) Proposition 3.4.64. Assume that (a n-1 , • • • , a 0 ) is n-generic.
Then we have

S k 1 ,w0 • S w -1 n-1 = κ (1) n S k 0 ,w0 and S k 1, ,w0 • S (w n-1 ) -1 = κ (2) n S k 0 ,w0 on the 1-dimensional space ( π • 0 ) I(1), µ0 for all 1 ≤ m ≤ n -1.
Proof. By the case w = w 0 of Proposition 3.4.60 and the fact that

k m m+1,m+2 = n -2 + [a 0 -a n-m-1 ] 1 and k m, n-m-1,n-m = n -2 + [a m -a n-1 ] 1 we have S k m ,w0 • S sn-m-1 = (-1) an-m-1 J(n -2 + [a 0 -a n-m-1 ] 1 , [a n-m-1 -a n-1 ] 1 ) S k m+1 ,w0 and S k m, ,w0 • S sm+1 = (-1) a0 J(n -2 + [a m -a n-1 ] 1 , [a 0 -a m ] 1 ) S k m+1, ,w0
on the 1-dimensional space (π • 0 ) I(1), µ0 for all 1 ≤ m ≤ n-2, and hence we finish the proof by induction on m. Lemma 3.4.65. We have

κ (1) n ≡ (-1) n-2 m=1 am n-2 m=1 (n-2+[a0-an-m-1]1)!([an-m-1-an-1]1)! (n-2+[a0-an-1]1)! (mod p) κ (2) n ≡ (-1) (n-2)a0 n-2 m=1 (n-2+[am-an-1]1)!([a0-am]1)! (n-2+[a0-an-1]1)! (mod p).
In particular,

ord p (κ (1) n ) = ord p (κ (2) n ) = 0.
Proof. This follows directly from (3.4.58), the definition of κ

(1)

n , κ (2)
n , and the fact that

(a n-1 , • • • , a 0 ) is n-generic. Corollary 3.4.66. Assume that (a n-1 , • • • , a 0 ) is n-generic.
Then we have

S k 1 ,w0 = p n-2 κ (1) n S k 0 ,w0 • S wn-1 and S k 1, ,w0 = p n-2 κ (2) n S k 0 ,w0 • S w n-1 on the 1-dimensional space ( π • 0 ) I(1), µ0 for all 1 ≤ m ≤ n -1. Proof. We have S w -1 n-1 • S wn-1 = p n-2 = S (w n-1 ) -1 • S w n-1 ,
so that this follows from Proposition 3.4.64 and Lemma 3.4.51.

Corollary 3.4.67. We have the equality

S k 1 ,w0 π U (Fp),µ1 0 = S k 1, ,w0 π U (Fp),µ 1 0 = S k 0 ,w0 π U (Fp),µ0 0
.

Proof. The follows directly from Proposition 3.4.60,

S 0,w -1 n-1 π U (Fp),µ0 0 = π U (Fp),µ1 0 and S 0,(w n-1 ) -1 π U (Fp),µ0 0 = π U (Fp),µ 1 0
and Lemma 3.4.65 by taking mod p reduction.

As in (3.4.34), we use the shorten notation S n := S k 1 ,w0 and S n := S k 1, ,w0 .

To state the main result in this section, we also define

P n := n-2 k=1 n-2 j=1 [a k -a n-1 ] 1 + j [a 0 -a k ] 1 + j = n-2 k=1 n-3 j=0 a k -a n-1 + j a 0 -a k + j ∈ Z × p , (3.4
.68)

ε * := n-2 m=1
(-1) a0-am , (3.4.69) and

κ n := κ (2) n (κ (1) n ) -1 . (3.4.70)
The main result of this section is the following theorem, which is a generalization of the case n = 3 in [HLM17], (3.2.1).

Theorem 3.4.71. Let Π n := Ind G(Qp)
B(Qp) χ be a tamely ramified principal series representation where the

χ = χ 1 ⊗ • • • ⊗ χ n : T (Q p ) → E × is a smooth character satisfying χ| T (Zp) = µ 1 .
On the 1-dimensional subspace Π I(1), µ1 n we have the identity:

S n • (Ξ n ) n-2 = p n-2 κ n n-2 k=1 χ k (p) S n for some κ n ∈ O × E (defined in (3.4.70)) such that κ n ≡ ε * P n (a n-1 , • • • , a 0 ) (mod E ) (3.4.72)
where ε * = ±1 is a sign defined in (3.4.69) and P n is a rational function defined in (3.4.68).

96CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
The following is a direct generalization of Lemma 3.2.5 in [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF].

Lemma 3.4.73. We have the equality

U r n = r k=1 χ k (p) -1
on the 1-dimensional space Π I(1), µ1 n

for each 1 ≤ r ≤ n -1.

Proof. The proof of this lemma is an immediate calculation which is parallel to that of [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], Lemma 3.2.5.

Proof of Theorem 3.4.71. Notice that

w n-1 (w * ) n-2 = w n-1 and (w n-1 ) + ((w * ) n-2 ) = 3(n -2) = (w n-1 ) + 2(n -2),
so that by Lemma 3.4.51 we have

S w n-1 • S (w * ) n-2 = p n-2 S wn-1 . (3.4.74)
By composing S k 0 ,w0 on both sides of (3.4.74), we deduce from Proposition 3.4.64 that

(κ (2) n ) -1 S n • S (w * ) n-2 = p n-2 (κ (1) n ) -1 S n and thus S n • S (w * ) n-2 = p n-2 κ n S n (3.4.75)
on the 1-dimensional subspace Π I(1), µ1 n

. Now Lemma 3.4.54 together with Lemma 3.4.73 gives rise to the identity in the statement of this theorem.

Finally, one can readily check from Lemma 3.4.65 that

κ n ≡ κ (2) n (κ (1) n ) -1 ≡ (-1) n-2 m=1 a0-am n-2 m=1 (n -2 + [a 0 -a n-m-1 ] 1 )!([a n-m-1 -a n-1 ] 1 )! (n -2 + [a m -a n-1 ] 1 )!([a 0 -a m ] 1 )! ≡ (-1) n-2 m=1 a0-am n-2 m=1 n-2 =1 + [a 0 -a m ] 1 + [a m -a n-1 ] 1 ≡ ε * P n (mod E ).
Note that ord p (κ n ) = 0. This completes the proof.

Special vectors in a dual Weyl module

Let R be a F p -algebra, and A ∈ G(R) a matrix. For J 1 , J 2 ⊆ {1, 2, • • • , n -1, n}, we write A J1,J2 for the submatrix of A consisting of the entries of A at the (i, j)-position for i ∈ J 1 , j ∈ J 2 . We define

J i 0 := {1, 2, • • • , i} ⊆ {1, • • • , n} for each 1 ≤ i ≤ n. We fix a tuple of integers (h 1 , • • • , h s ) for some 1 ≤ s ≤ n -1 such that 1 ≤ h r ≤ n -1 for all 1 ≤ r ≤ s and s r=1 h r = n -1
and we denote this tuple by h. Then we can define n -1 positive roots β h,i for 1 ≤ i ≤ n -1 as follows. Given an integer 2 ≤ i ≤ n -1, there exists an unique integer 1 ≤ r 0 ≤ s -1 such that

r0 r=1 h r < i ≤ r0+1 r=1 h r ,
and we use the notation

[i] h := r0 r=1 h r .
Then we define

β h,i := i k=1+[i] h α k .
We always set β h,1 = α 1 .

Then we define Φ

+ h := {α ∈ Φ + | α = β h,i
for all 1 ≤ i ≤ n -1} and notice that this set gives an unipotent group U h U by setting

U h := α∈Φ + h U α .
We emphasize that all U h constructed here are good in the sense of Definition 3.4.21. In particular, if s = n -1 and n r = 1 for 1 ≤ r ≤ n -1 we recover [U, U ], and if s = 1 and n 1 = n -1 we recover U 1 (c.f. Example 3.4.23). We define U h as the reduction of U h mod p.

Given a tuple h as above, we define the subsets

J i h ⊆ {1, • • • , n} for 1 ≤ i ≤ n -1 as J i h := {1, 2, • • • , i + 1} \ { r 0 r=1 h r } if r 0 r=1 h r ≤ i < r 0 +1 r=1 h r
for some 1 ≤ r 0 ≤ s -1 and we use the notation

[i] h := r 0 r=1 h r .
It is easy to see that

|J i h | = i for 1 ≤ i ≤ n -1. We define D h,i := det (w 0 A) J i 0 ,J i h for all 1 ≤ i ≤ n-1. We also set D n := det(w 0 A). Hence, D h,i (1 ≤ i ≤ n-1)
O(G) U := {f ∈ O(G) | f • u = f ∀u ∈ U } = ⊕ λ∈X(T ) O(λ).
(3.4.76)

It follows from the definition of the dual Weyl module as an algebraic induction that we have a natural isomorphism H 0 (λ) ∼ = O(λ).

(3.4.77)

Note by [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF], Proposition II.2.6 that H 0 (λ) = 0 if and only if λ ∈ X(T ) + .

We often write the weight λ explicitly as

(d 1 , d 2 , • • • , d n ) where d i ∈ Z for 1 ≤ i ≤ n.
We will restrict our attention to a p-restricted and dominant λ, i.e.,

d 1 ≥ d 2 ≥ ... ≥ d n and d i-1 -d i < p for 2 ≤ i ≤ n.
We recall from the beginning of Section 3.4 the notation (•) λ for a weight space with respect to the weight λ . We define Σ to be the set of (n -1)-tuple of integers m = (m 1 , ..., m n-1 ) satisfying 0

≤ m i ≤ d i -d i+1 for 1 ≤ i ≤ n -1.
For each tuple m, we can define a vector

v alg h,m := D dn n n-1 i=1 D di-di+1-mi i (D h,i ) mi . Proposition 3.4.78. Let λ = (d 1 , d 2 , • • • , d n ) ∈ X 1 (T ). The set {v alg h,m | m ∈ Σ} (3.4.79)
forms a basis of H 0 (λ) U h . Moreover, the weight of v alg h,m is

λ - n-1 i=1 m i β h,i
and thus each element in (3.4.79) has distinct weight.

Proof. We define

U h O(G) U := {f ∈ O(G) | u 1 • f = f • u = f ∀u ∈ U & ∀u 1 ∈ U h } and U h M(G) U := {f ∈ M(G) | u 1 • f = f • u = f ∀u ∈ U & ∀u 1 ∈ U h }.
We consider a matrix A such that its entries A i,j are indefinite variables. Then we can formally do Bruhat decomposition

A = U A w 0 T A,h U A,h
such that the entries of U A , T A,h , U A,h are rational functions of A i,j satisfying

(U A ) i,j = 1 if i = j; 0 if i > j, (T A,h ) i,j =    D i (A) if i = j; D h,k (A) if (i, j) = β h,k ; 0 otherwise , (U A,h ) i,j = 1 if i = j; 0 if i > j or (i, j) = β h,k for some 1 ≤ k ≤ n -1.
For each rational function f ∈ U h M(G) U , we notice that f only depends on T A,h , which means that f is rational function of D i for 1 ≤ i ≤ n and D h,i for 1 ≤ i ≤ n -1. In other word, we have

U h M(G) U = F p D 1 , • • • , D n , D h,1 , • • • , D h,n-1 ⊆ M(G).

Then we define

U h ,λ O(G) U ,λ := {f ∈ U h O(G) U | x • f = λ (x)f, and f • x = λ(x)f ∀x ∈ T } and U h ,λ M(G) U ,λ := {f ∈ U h M(G) U | x • f = λ (x)f, and f • x = λ(x)f ∀x ∈ T }.

Note that we have and an obvious inclusion

U h ,λ O(G) U ,λ ⊆ U h ,λ M(G) U ,λ .
We can also identify U h ,λ O(G) U ,λ with H 0 (λ)

U h
λ via the isomorphism (3.4.77). By definition of D i (resp. D h,i ) we know that they are T -eigenvector with eigencharacter

i k=1 k (resp. ( i+1 k=1 k )-[i] h ) for 1 ≤ i ≤ n (resp. for 1 ≤ i ≤ n -1
). Therefore we observe that U h ,λ M(G) U ,λ is one dimensional for any λ, λ ∈ X(T ) and is spanned by

D dn n n-1 i=1 D di-di+1-mi i (D h,i ) mi where λ = (d 1 , • • • , d n ) and λ = λ - n-1 i=1 m i β h,i .
As O(G) is a UFD and D i , D h,i are irreducible, we deduce that

D dn n n-1 i=1 D di-di+1-mi i (D h,i ) mi ∈ O(G) if and only if 0 ≤ m i ≤ d i -d i+1 for all 1 ≤ i ≤ n -1 if and only if H 0 (λ) U h λ = 0 which finishes the proof.
We consider the special case of Proposition 3.4.78 when s = 1 and n 1 = n -1.

Corollary 3.4.80. Let λ = (d 1 , d 2 , • • • , d n ) ∈ X 1 (T ). For λ ∈ X(T ), we have dim Fp H 0 (λ) U 1 λ ≤ 1.
Moreover, the set of λ such that the space above is nontrivial is described explicitly as follows: consider the set Σ {n-1} of (n -1)-tuple of integers m = (m 1 , ..., m n-1 ) satisfying m i ≤ d i -d i+1 for 1 ≤ i ≤ n -1, and let

v alg {n-1},m := D dn n n-1 i=1 D di-di+1-mi i (D {n-1},i ) mi . 100CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Then the set {v alg {n-1},m | m ∈ Σ {n-1} }
forms a basis of the space H 0 (λ) U 1 , and the weight of the vector v alg {n-1},m is

(d 1 - n-1 i=1 m i , d 2 + m 1 , ..., d n-1 + m n-2 , d n + m n-1 ).
Remark 3.4.81. Corollary 3.4.80 essentially describes the decomposition of an irreducible algebraic representation of GL n after restricting to a maximal Levi subgroup which is isomorphic to GL 1 × GL n-1 . This classical result is crucial in the proof of Theorem 3.4.137.

Some technical formula

In this section, we prove a technical formula that will be used in Section 3.4.8. The main result of this section is Proposition 3.4.101.

From now on, we assume that (a n-1 , • • • , a 0 ) is n-generic in the lowest alcove (c.f. Definition 3.4.5). We need to do some elementary calculation of Jacobi sums. For this purpose we need to define the following group operators for 2 ≤ r ≤ n -1:

X + r := t∈Fp t p-2 u n-1 i=r αi (t) ∈ F p [G(F p )],
and similarly

X - r := t∈Fp t p-2 w 0 u n-1 i=r αi (t)w 0 ∈ F p [G(F p )].
We notice that by definition we have the identification X + r = X n-1 i=r αi,1 , where X n-1 i=r αi,1 is defined in (3.4.8).

Lemma 3.4.82. For a tuple of integers k

= (k i,j ) ∈ {0, 1, • • • , p -1} |Φ + w 0 | , we have X + r • S k,w0 = k r,n S k r,n,+ ,w0
where k r,n,+ = (k r,n,+ i,j

) satisfies k r,n,+ r,n = k r,n -1, and k r,n,+ i,j

= k i,j if (i, j) = (r, n).
Proof. This is just a special case of Lemma 3.4.11 when α 0 = n i=r α i and m = 1.

For the following lemma, we set

I := {(i 1 , i 2 , • • • , i s ) | 1 ≤ i 1 < i 2 < • • • < i s = n for some 1 ≤ s ≤ n -1}.
to lighten the notation.

Lemma 3.4.83. Let X = (X i,j ) 1≤i,j≤n be a matrix satisfying

X i,j = 0 if 1 ≤ j < i ≤ n -1.
Then the determinant of X is

det(X) = (i1,••• ,is)∈I (-1) s-1 X n,i1   j =i k , 1≤k≤s X j,j   s-1 k=1 X i k ,i k+1 . (3.4.84)
Proof. By definition of the determinant we know that

det(X) = w∈W (-1) (w) n k=1 X k,w(k) .
From the assumption on X, we know that each w that appears in the sum satisfies

w(k) < k (3.4.85) for all 2 ≤ k ≤ n -1.
Assume that w has the decomposition into disjoint cycles

w = (i 1 1 , i 1 2 , • • • , i 1 n1 ) • • • (i m 1 , i m 2 , • • • , i m nm ) (3.4.86)
where m is the number of disjoint cycles and n k ≥ 2 is the length for the k-th cycle appearing in the decomposition.

We observe that the largest integer in {i k j | 1 ≤ j ≤ n k } must be n for each 1 ≤ k ≤ m by condition (3.4.85). Therefore we must have m = 1 and we can assume without loss of generality that i 1 n1 = n. It follows from the condition (3.4.85) that

i 1 j < i 1 j+1
for all 1 ≤ j ≤ n 1 -1. Hence we can set

s := n 1 , i 1 := i 1 1 , • • • , i s := i 1 n1 .
We observe that (w) = s -1 and the formula (3.4.84) follows.

Recall from the beginning of Section 3.4.7 that we use the notation A J1,J2 for the submatrix of A consisting of the entries at the (i, j)-position with i ∈ J 1 , j ∈ J 2 , where J 1 , J 2 are two subsets of {1, 2, • • • , n} with the same cardinality. For a pair of integers (m, r)

with 1 ≤ m ≤ r -1 ≤ n -2, we let J m,r 0 := {1, 2, • • • , r, n -m + 1}.
For a matrix A ∈ U (F p ), an element t ∈ F p , and a triple of integers (m, r, ) satisfying 1 ≤ m ≤ r -1 ≤ n -2 and 1 ≤ ≤ n -1, we define some polynomials as follows:

     D m,r (A, t) := det u n-1 i=r αi (t)w 0 Aw 0 J m,r 0 ,J n-r+1 0 when 1 ≤ m ≤ r -1; D ( ) r (A, t) := det u n-1 i=r αi (t)w 0 Aw 0 J 0 ,J 0 when 1 ≤ ≤ n -r (3.4.87)
We define the following subsets of I: for each 1 ≤ ≤ n -1

I := {(i 1 , i 2 , • • • , i s ) ∈ I | n -+ 1 ≤ i 1 < i 2 < • • • < i s = n for some 1 ≤ s ≤ }.
Note that we have natural inclusions

I ⊆ I ⊆ I if 1 ≤ ≤ ≤ n -1.
In particular, I 1 has a unique element (n). Similarly, for each 1 ≤ ≤ n -1 we define

I := {(i 1 , i 2 , • • • , i s ) | 1 ≤ i 1 < i 2 < • • • < i s-1 ≤ n -< i s = n for some 1 ≤ s ≤ },
and we set

I := I ∩ I 102CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
for all 1 ≤ ≤ -1 ≤ n -2. We often write i = (i 1 , • • • , i s ) for an arbitrary element of I, and define the sign of i by ε(i) := (-1) s .

We emphasize that all the matrices w 0 u n-1 i=r αi (t)w 0 Aw 0

J m,r 1 ,J n-r+1
2 for 1 ≤ m ≤ r -1, and all the matrices w 0 u n-1 i=r αi (t)w 0 Aw 0

J 1 ,J 2
for 1 ≤ ≤ n -r, after multiplying a permutation matrix, satisfy the conditions on the matrix X in Lemma 3.4.83. Hence, by Lemma 3.4.83 we notice that

D m,r (A, t) = A m,r + tf m,r (A) when 1 ≤ m ≤ r -1; D ( ) r (A, t) = 1 -tf r,n-+1 (A) wehn 1 ≤ ≤ n -r; (3.4.88)
where for all 1 ≤ m ≤ r -1

f m,r (A) := i∈In-r+1   ε(i)A m,i1 s j=2 A ij-1,ij   . (3.4.89) Let (m, r) be a tuple of integers with 1 ≤ m ≤ r -1 ≤ n -2. Given a tuple of integers k ∈ {0, 1, • • • , p -1} |Φ + w 0 | , i = (i 1 , i 2 , • • • , i s ) ∈ I n-r+1
, and an integer r satisfying 1 ≤ r ≤ r, we define two tuples of integers

k i,m,r = (k i,m,r i,j ) ∈ {0, 1, • • • , p -1} |Φ + w 0 | and k i,m,r,r = (k i,m,r,r i,j ) ∈ {0, 1, • • • , p -1} |Φ + w 0 | as follows: k i,m,r i,j =            k m,r -1 if (i, j) = (m, r) and i 1 > r; k m,i1 + 1 if (i, j) = (m, i 1 ) and i 1 > r; k m,r if (i, j) = (m, r) and i 1 = r; k i,j + 1 if (i, j) = (i h , i h+1 ) for 1 ≤ h ≤ s -1; k i,j otherwise and k i,m,r,r i,j = k i,m,r r ,n -1 if (i, j) = (r , n); k i,m,r i,j otherwise.
Finally, we define one more tuple of integers k

r,+ = (k r,+ i,j ) ∈ {0, 1, • • • , p -1} |Φ + w 0 | by k r,+ i,j := k r,n + 1 if (i, j) = (r, n); k i,j otherwise.
Lemma 3.4.90. Fix two integers r and m such that

1 ≤ m ≤ r -1 ≤ n -2, and let k = (k i,j ) ∈ {0, 1, • • • , p -1} |Φ + w 0
| . Assume that k i,j = 0 for r + 1 ≤ j ≤ n -1 and that k i,r = 0 for i = m, and assume further that

a n-r -a 1 + [a 1 -a n-1 - n-1 i=1 k i,n ] 1 + k m,r < p.
Then we have

X - r • S k,w0 v 0 = k m,r i∈In-r ε(i)S k i,m,r ,w0 v 0 + ([a n-r -a n-1 - n-1 i=1 k i,n ] 1 + k m,r )S k r,+ ,w0 v 0 - n-r =2 (a n-r -a -1 + k m,r )   i∈I \I -1 ε(i)S k i,r,n-+1 ,w0 v 0   .
Proof. By the definition of X - r , we have

X - r • S k,w0 v 0 = A∈U (Fp),t∈Fp   t p-2   1≤i<j≤n A ki,j i,j   w 0 u n-1 h=r α h (t)w 0 Aw 0   v 0 . (3.4.91)
For an element w ∈ W , we use E w to denote the subset of

U (F p ) × F p consisting of all (A, t) such that w 0 u n-1 h=r α h (t)w 0 Aw 0 ∈ B(F p )wB(F p ). It is not difficult to see that if E w = ∅ then ww 0 (i) = i for all 1 ≤ i ≤ r -1.
We define M w to be

M w := (A,t)∈Ew   t p-2   1≤i<j≤n A ki,j i,j   w 0 u n-1 h=r α h (t)w 0 Aw 0   v 0 .
By the definition of E w , we deduce that there exist A ∈ U w (F p ), A ∈ U (F p ), and T ∈ T (F p ) for each given (A, t) ∈ E w such that w 0 u n-1 h=r α h (t)w 0 Aw 0 = A wT A .

(3.4.92)

Here, the entries of A , T and A are rational functions of t and the entries of A. We can rewrite (3.4.92) as

w 0 u n-1 h=r α h (-t)w 0 A w = Aw 0 T -1 (T (A ) -1 T -1 ) (3.4.93)
In other words, the right side of (3.4.93) can also be viewed as the Bruhat decomposition of the left side. Therefore the entries of A, T , A can also be expressed as rational functions of the entries of A .

For each A ∈ U w (F p ) and w ∈ W , we define

       D w m,r (A , t) := det u n-1 i=r αi (t)w 0 A w J m,r 0 ,J n-r+1 0 when 1 ≤ m ≤ r -1; D w,( ) r (A , t) := det u n-1 i=r αi (t)w 0 A w J 0 ,J 0 when 1 ≤ ≤ n -r.
(3.4.94)

Note that if w = w 0 , then the definition in (3.4.94) specializes to (3.4.87). We notice that for a given matrix A ∈ U w (F p ), the equality (3.4.93) exists if and only if

D w,( ) r (A , -t) = 0 for all 1 ≤ ≤ n -r. (3.4.95)
On the other hand, we also notice that given a matrix A ∈ U (F p ), the equality (3.4.93) exists if and only if (3.4.95) holds.

T -1 = diag D w, (1) r , D w,(2) r D w,(1) r
, • • • , D w,(n-r) r D w,(n-1-r) r , 1 D w,(n-r) r , 1, • • • , 1 (3.4.96)
in which we write D w,(i) r for D w,(i) r

(A , -t) for brevity. We also have

A i,j =      A i,j if 1 ≤ i < j ≤ n and j ≤ r -1; D w m,r (A , -t) if (i, j) = (m, r); A i,n D w, (1) r
(A ,-t)

if 1 ≤ i ≤ n -1 and j = n.

(3.4.97)

We apply (3.4.92), (3.4.97) and (3.4.96) to M w and get

M w = (A,t)∈Ew     F (A , w, t)     1≤i<j≤n j≤r or j=n (A i,j ) ki,j     A w 0     v 0 (3.4.98)
where

F (A , w, t) := t p-2 (D w m,r ) km,r (D w,(1) r ) a1-an-1-n-1 i=1 ki,n n-r s=2 (D w, (s) r
) as-as-1

in which we let D w m,r := D w m,r (A , -

(A , -t) for brevity. We have discussed in (3.4.95) that (A, t) ∈ E w is equivalent to (A , t) ∈ U w (F p ) × F p satisfying the conditions in (3.4.95). As each D w,(s) r (A , -t) appears in F (A , w, t) with a positive power, we can automatically drop the condition (3.4.95) and reach

M w = (A,t)∈U w (F0)×Fp     F (A , w, t)     1≤i<j≤n j≤r or j=n (A i,j ) ki,j     A w 0     v 0 .
(3.4.99) If w = w 0 , then there exist a unique integer i 0 satisfying r ≤ i 0 ≤ n such that ww 0 (i 0 ) < i 0 but ww 0 (i) = i for all i 0 + 1 ≤ i ≤ n.

By applying Lemma 3.4.83 to D w,(n+1-i0) r

(A , -t) (as (u n-1 i=r αi (t)w 0 A w) J 0 ,J 0 satisfy the condition of Lemma 3.4.83 after multiplying a permutation matrix), we deduce that

D w,(n+1-i0) r (A , -t) = tf (A )
where f (A ) is certain polynomial of entries of A . Now we consider F (A , w, t) as a polynomial of t. The minimal degree of monomials of t appearing in

F (A , w, t) is at least p -2 + a n+1-i0 -a n-i0 > p -1,
and the maximal degree of monomials of t appearing in F (A , w, t) is

p -2 + k m,r + [a 1 -a n-1 - n-1 i=1 k i,n ] 1 + n-r s=2 a s -a s-1 = p -2 + k m,r + [a 1 -a n-1 - n-1 i=1 k i,n ] 1 + a n-r -a 1 < 2(p -1).
As a result, the degree of each monomials of t in F (A , w, t) is not divisible by p -1. Hence, we conclude that M w = 0 for all w = w 0 as we know that t∈Fp t k = 0 if and only if p -1 | k and k = 0. Finally, we calculate M w0 explicitly using (3.4.99). Indeed, by applying (3.4.88), the monomials of t appearing in F (A , w 0 , t) is nothing else than

t p-1 -k m,r f m,r (A ) + [a 1 -a n-1 - n-1 i=1 k i,n ] 1 f r,n (A ) + n-r s=2 (a s -a s-1 )f r,n+1-s (A ) .
As t∈Fp t p-1 = -1, we conclude that

X - r • S k,w0 v 0 = M w0 = A ∈U (Fp)     F 0 (A )     1≤i<j≤n j≤r or j=n (A i,j ) ki,j     A w 0     v 0 (3.4.100)
where

F 0 (A ) := k m,r f m,r (A ) -[a 1 -a n-1 - n-1 i=1 k i,n ] 1 f r,n (A ) - n-r s=2 (a s -a s-1 )f r,n+1-s (A ).
Recalling the explicit formula of f m,r and f r,n+1-s for 1 ≤ s ≤ n -r from (3.4.89) and then rewriting (3.4.100) as a sum of distinct monomials of entries of A finishes the proof.

Proposition 3.4.101. Keep the assumptions and the notation of Lemma 3.4.90.

Then we have

X + r • X - r • S k,w0 v 0 = k m,r k r,n i∈In-r ε(i)S k i,m,r ,w0 v 0 + (k r,n + 1) [a n-r -a n-1 - n-1 i=1 k i,n ] 1 + k m,r S k r,+ ,w0 v 0 -k r,n n-r =2 (a n-r -a -1 + k m,r )   i∈I \I -1 ε(i)S k i,r,n-+1 ,w0 v 0   .
Proof. This is just a direct combination of Lemma 3.4.90 and Lemma 3.4.82.

Proof of Theorem 3.4.35

The main target of this section is to prove Theorem 3.4.137. This theorem together with Corollary 3.4.67 immediately implies Theorem 3.4.35. We start this section by introducing some notation. We first define a subset Λ w0 of {0,

• • • , p -1} |Φ + w 0 | consisting of the tuples k = (k i,j ) i,j such that for each 1 ≤ r ≤ n -1 1≤i≤r<j≤n k i,j = [a 0 -a n-1 ] 1 + n -2.
Note that the set Λ w0 embeds into π 0 by sending k to S k,w0 v 0 . Moreover, this family of vectors {S k,w0 v 0 | k ∈ Λ w0 } shares the same eigencharacter by Lemma 3.4.19.

We define k ∈ Λ w0 where k = (k i,j ) is defined by k 1,n = [a 0 -a n-1 ] 1 + n -2 and k i,j = 0 otherwise. We define V to be the subrepresentation of π 0 generated by S k ,w0 v 0 . We also need to define several useful elements and subsets of Λ w0 . For each 1 ≤ r ≤ n -1, we define k ,r ∈ Λ w0 where k ,r = (k ,r i,j ) is defined by

k ,r i,j :=    n -2 + [a 0 -a n-1 ] 1 if 2 ≤ j = i + 1 ≤ r; n -2 + [a 0 -a n-1 ] 1 if (i, j) = (r, n); 0 otherwise.
In particular, we have

k ,1 = k and k ,n-1 = k 0 (3.4.102)
where k 0 is defined in (3.4.33).

For each 1 ≤ r ≤ n -2 and 0 ≤ s ≤ [a 0 -a n-1 ] 1 + n -2, we define a tuple k ,r,s ∈ Λ w0 as follows:

k ,r,s i,j =            n -2 + [a 0 -a n-1 ] if 2 ≤ j = i + 1 ≤ r; n -2 + [a 0 -a n-1 ] 1 -s if (i, j) = (r, r + 1); s if (i, j) = (r, n); n -2 + [a 0 -a n-1 ] 1 -s if (i, j) = (r + 1, n); 0 otherwise.
In particular, we have k ,r,0 = k ,r+1 and k ,r,[a0-an-1]1+n-2 = k ,r (3.4.103)

for each 1 ≤ r ≤ n -2. We now introduce the rough idea of the proof of Theorem 3.4.137. The first obstacle to generalize the method of Proposition 3.1.2 in [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] is that V 0 does not admit a structure as G-representation in general. Our method to resolve this difficulty is to replace S k 0 ,w0 v 0 by S k ,w0 v 0 . We prove in Proposition 3.4.133 that V admits a structure as G-representation and actually can be identified with a dual Weyl module H 0 (µ w0 0 ). (The notation µ w0 0 will be clear in the following.) Now it remains to prove that S k ,w0 v 0 ∈ V 0 (3.4.104) to deduce Theorem 3.4.137. We will prove in Proposition 3.4.124 that

S k ,r,s-1 ,w0 v 0 ∈ V 0 =⇒ S k ,r,s ,w0 v 0 ∈ V 0 for each 1 ≤ r ≤ n -2 and 1 ≤ s ≤ [a 0 -a n-1 ] 1 + n -2.
As a result, we can thus pass from

S k 0 ,w0 v 0 ∈ V 0 to S k ,r ,w0 v 0 ∈ V 0 for r = n -2, n -3, • • • , 1. The identification k = k ,1 thus gives us (3.4.104).
We firstly state three direct Corollaries of Proposition 3.4.101. It is easy to check that each tuple k appearing in the following Corollaries satisfies the assumption in Proposition 3.4.101.

Corollary 3.4.105. For each 2 ≤ r ≤ n -1 and 0 ≤ s ≤ [a 0 -a n-1 ] 1 + n -3, we have X + r • X - r • S k ,r-1,s ,w0 v 0 = ([a 0 -a n-1 ] 1 + n -2 -s) 2 i∈In-r ε(i)S (k ,r-1,s ) i,m,r,r ,w0 v 0 + ([a n-r -a n-1 ] 1 -s)([a 0 -a n-1 ] 1 + n -1 -s)S k r-1,s ,w0 v 0 -([a 0 -a n-1 ] 1 + n -2 -s) n-r =2 (a n-r -a -1 + [a 0 -a n-1 ] 1 + n -2 -s) •   i∈I \I -1 ε(i)S (k ,r-1,s ) i,r,n-+1,r ,w0 v 0   .
Corollary 3.4.106. Fix two integers r and m such that 1 ≤ m ≤ r -1 ≤ n -2, and let k = (k i,j ) be a tuple of integers in Λ w0 such that

k i,j =            0 if r + 1 ≤ j ≤ n -1; 0 if i = m and j = r; 0 if r + 1 ≤ i ≤ n -1 and j = n; 1 if (i, j) = (m, r); 1 if (i, j) = (r, n).
Then we have

X + r • X - r • S k,w0 v 0 = i∈In-r ε(i)S k i,m,r,r ,w0 v 0 + 2(a n-r -a 0 -n + 3)S k,w0 v 0 - n-r =2 (a n-r -a -1 + 1)   i∈I \I -1 ε(i)S k i,r,n-+1,r ,w0 v 0   .
Corollary 3.4.107. Fix two integers r and m such that 1 ≤ m ≤ r -1 ≤ n -2, and let k = (k i,j ) be a tuple of integers in Λ w0 such that

k i,j = 0 if r ≤ j ≤ n -1; 0 if r ≤ i ≤ n -1 and j = n.
Then we have

X + r • X - r • S k,w0 v 0 = (a n-r -a 0 -n + 2)S k,w0 v 0 - n-r =2 (a n-r -a -1 + 1)   i∈I \I -1 ε(i)S k i,r,n-+1,r ,w0 v 0   .
We now define the following constants in F p :

c := -1 k=1 (a k -a 0 -n + 2 + k) 2 -k-1 ; c := (a -a 0 -n + 3 + )c
for all 1 ≤ ≤ n -1 where we understand c 1 to be 1. As the tuple (a n-1 , • • • , a 0 ) is n-generic in the lowest alcove, we notice that c = 0 = c for all 1 ≤ ≤ n -1. By definition of c k and c k one can also easily check that

-1 k=1 (c k -c k ) = c . (3.4.108)
We also define inductively the constants: for each 1 ≤ ≤ n -1

d , := 2(a -a 0 -n + 3) if = 0; c d , -1 -(a -a + 1)c -1 k=1 (c k -c k ) if 1 ≤ ≤ -1.
We further define inductively a sequence of group operators Z as follows:

Z 1 := d 1,0 Id -X + n-1 • X - n-1 ∈ F p [G(F p )]
and

Z := d , -1 Id -Z -1 • • • • • Z 1 • X + n-• X - n- ∈ F p [G(F p )]
for each 2 ≤ ≤ n -2.

108CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Lemma 3.4.109. For 1 ≤ ≤ n -1, we have the identity

d , -1 = (a -a 0 -n + 2) -1 k=1 c k + c .
Proof. During the proof of this lemma, we will keep using the following obvious identity with two variables ab = (a + 1)(b -1) + a -b + 1 (3.4.110)

By definition of d , -1 we know that

d , -1 = 2(a -a 0 -n + 3) -1 k=1 c k - -1 =1   (a -a + 1)c   -1 k=1 (c k -c k )   -1 k= +1 c k   and therefore d , -1 -(a -a 0 -n + 2) -1 k=1 c k = (a -a 0 -n + 4) -1 k=1 c k - -1 =1   (a -a + 1)c   -1 k=1 (c k -c k )   -1 k= +1 c k   .
Now we prove inductively that for each 1

≤ j ≤ -1 d , -1 -(a -a 0 -n + 2) -1 k=1 c k = (a -a 0 -n + 3 + j) j-1 k=1 (c k -c k )   -1 k=j c k   - -1 =j   (a -a + 1)c   -1 k=1 (c k -c k )   -1 k= +1 c k   . (3.4.111)
By the identity (3.4.110), one can easily deduce that

(a -a 0 -n + 3 + j)c j -(a -a j + 1)c j = [(a -a 0 -n + 3 + j)(a j -a 0 -n + 3 + j) -(a -a j + 1)] c j = (a -a 0 -n + 4 + j)(a j -a 0 -n + 2 + j)c j = (a -a 0 -n + 4 + j)(c j -c j ).
Hence, we get the identity:

(a -a 0 -n + 3 + j)c j -(a -a j + 1)c j   -1 k=j+1 c k   j-1 k=1 (c k -c k ) = (a -a 0 -n + 4 + j) j k=1 (c k -c k )   -1 k=j+1 c k   . (3.4.112)
Thus, if the equation (3.4.111) holds for j, we can deduce that it also holds for j + 1. By taking j = -1 and using the equation (3.4.112) once more, we can deduce that

d , -1 -(a -a 0 -n + 2) -1 k=1 c k = (a -a 0 -n + 3 + ) -1 k=1 (c k -c k ) .
Hence, by the equation (3.4.108), one finishes the proof. Proposition 3.4.113. Fix two integers r and m such that 1 ≤ m ≤ r -1 ≤ n -2.

(i) Let k = (k i,j ) be as in Corollary 3.4.106. Then we have

Z n-r • S k,w0 = c n-r S k ,w0 (3.4.114)
where k = (k i,j ) is defined as follows:

k i,j :=    0 if (i, j) = (m, r) or (i, j) = (r, n); 1 if (i, j) = (m, n); k i,j otherwise.
(ii) Let k = (k i,j ) be as in Corollary 3.4.107. Then we have

Z n-r • S k,w0 = c n-r S k,w0 .
(3.4.115)

We prove this proposition by a series of lemmas.

Lemma 3.4.116. Proposition 3.4.113 is true for r = n -1.

Proof. For part (i) of Proposition 3.4.113, by applying Corollary 3.4.106 to the case r = n -1 we deduce that

X + n-1 • X - n-1 • S k,w0 v 0 = 2(a 1 -a 0 -n + 3)S k,w0 v 0 -S k i 0 ,m,n-1,n-1 ,w0 v 0
where i 0 = {n -1, n}. Hence, part (i) of the proposition follows directly from the definition of Z 1 and c 1 .

For part (ii) of Proposition 3.4.113, again by Corollary 3.4.107 to the case r = n -1 we deduce that

X + n-1 • X - n-1 • S k,w0 v 0 = (a 1 -a 0 -n + 2)S k,w0 v 0 . Then we have Z 1 • S k,w0 v 0 = (a 1 -a 0 -n + 4)S k,w0 v 0
and part (ii) of the proposition follows directly from the definition of c 1 .

Lemma 3.4.117. Let be an integer with 2 ≤ ≤ n-1. If Proposition 3.4.113 is true for r ≥ n-+1, then it is true for r = n -.

Proof. We prove part (ii) first. Assume that (3.4.115) holds for r ≥ n -+ 1. In fact, for a Jacobi sum S k,w0 satisfying the conditions in the (3.4.115) for r = n -, we have

X + n-• X - n-• S k,w0 v 0 = (a -a 0 -n + 2)S k,w0 v 0
by Corollary 3.4.107. Then we can deduce

Z -1 • • • • • Z 1 • X + n-• X - n-• S k,w0 v 0 = (a -a 0 -n + 2) -1 s=1 c s S k,w0 v 0
from the assumption of the Lemma. Hence, by definition of Z , we have

Z • S k,w0 v 0 = d , -1 S k,w0 v 0 -Z -1 • • • • • Z 1 • X + n-• X - n-• S k,w0 v 0 = d , -1 -(a -a 0 -n + 2) -1 s=1 c s S k,w0 v 0 = c S k,w0 v 0 110CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
where the last equality follows from Lemma 3.4.109. Now we turn to part (i). Assume that (3.4.114) holds for r ≥ n -+ 1. We will prove inductively that for each satisfying 1 ≤ ≤ -1, we have

Z • • • • • Z 1 • X + n-• X - n-• S k,w0 v 0 = d , S k,w0 v 0 +   s=1 (c s -c s )     i∈I ε(i)S k i,m,n-,n-,w0 v 0   +   s=1 (c s -c s )      -1 h= +1
(a -a h + 1)

i∈I h \I h+1 ε(i)S k i,n-,n-h,n-,w0 v 0    (3.4.118)
We begin with studying some basic properties of the index sets I h . First of all, the set I +1 \ I +2 has a unique element, which is precisely i = {n --1, n}. Furthermore, there is a natural map of sets res : I h → I +1 h for all + 2 ≤ h ≤ defined by eliminating the element n -from i ∈ I h if n -∈ i. In other words, for each i ∈ I +1 h , we have

res -1 ({i}) = {i, i ∪ {n -}} ⊆ I h .
We use the shorten notation i := i ∪ {n -}

for each i ∈ I +1 h . Note in particular that ε(i) = -ε(i ). Given an arbitrary i ∈ I +1 h for + 2 ≤ h ≤ -1, then S k i,n-,n-h,n-,w0 (resp. S k i ,n-,n-h,n-,w0
) satisfies the conditions before the equation (3.4.115) (resp. (3.4.114)). As a result, by the assumption that Proposition 3.4.113 is true for r = n --1, we deduce that

Z +1 • S k i,n-,n-h,n-,w0 v 0 -S k i ,n-,n-h,n-,w0 v 0 = c +1 -c +1 S k i,n-,n-h,n-,w0 v 0 . (3.4.119)
Similarly, we have

Z +1 • S k i,m,n-,n-,w0 v 0 -S k i ,m-,n-,w0 v 0 = c +1 -c +1 S k i,m,n-,n-,w0 v 0 (3.4.120)
for each i ∈ I +1 . We also have

Z +1 • S k,w0 v 0 = c +1 S k,w0 v 0 (3.
Z +1 • S k i 0 ,n-,n--1,n-,w0 v 0 = c +1 S k,w0 v 0 (3.
Z +1 • • • • • Z 1 • X + n-• X - n-• S k,w0 v 0 = d , Z +1 • S k,w0 v 0 +   s=1 (c s -c s )   Z +1 •   i∈I ε(i)S k i,m,n-,n-,w0 v 0   +   s=1 (c s -c s )   Z +1 •    -1 h= +1
(a -a h + 1)

i∈I h \I h+1 ε(i)S k i,n-,n-h,n-,w0 v 0   
which is the same as

c d , S k,w0 v 0 +   s=1 (c s -c s )   (X + Y + Z) (3.4.123) where X = (a -a + 1)Z +1 • S k i 0 ,n-,n--1,n-,w0 v 0 , Y = i∈I +1 ε(i)Z +1 • S k i,m,n-,n-,w0 v 0 -S k i ,m,n-,n-,w0 v 0 , and
Z = -1 h= +2
(a -a h + 1)

i∈I +1 h \I +1 h+1 ε(i)Z +1 • S k i,n-,n-h,n-,w0 v 0 -S k i ,n-,n-h,n-,w0 v 0 .
One can also readily check that (3.4.123) is also the same as

  c +1 d , + c +1   s=1 (c s -c s )   (a -a + 1)   S k,w0 v 0 +   +1 s=1 (c s -c s )      i∈I +1 ε(i)S k i,m,n-,n-,w0 v 0    +   +1 s=1 (c s -c s )      -1 h= +2
(a -a h + 1)

i∈I +1 h \I +1 h+1 ε(i)S k i,n-,n-h,n-,w0 v 0    ,
which finishes the proof of (3.4.118), as we have

d , +1 = c +1 d , + c +1   s=1 (c s -c s )   (a -a + 1)
by definition. Note that (3.4.118) for each 1 ≤ ≤ -1 then follows from Corollary 3.4.106 for r = n -. Note that the case = -1 for (3.4.118) is just the following

Z -1 • • • • • Z 1 • X + n-• X - n-• S k,w0 v 0 = d , -1 S k,w0 v 0 - -1 s=1 (c s -c s ) S k i 1 ,m,n-,n-,
≤ s ≤ [a 0 -a n-1 ] 1 + n -2, if S k ,r,s-1 ,w0 v 0 ∈ V 0 , then S k ,r,s ,w0 v 0 ∈ V 0 .
112CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Proof. By Proposition 3.4.113 and its proof, we can deduce the following equalities

Z n-2-r • • • • • Z 1 • S k ,r,s-1 ,w0 v 0 = n-2-r =1 c S k ,r,s-1 ,w0 v 0 , Z n-2-r • • • • • Z 1 •   i∈In-1-r ε(i)S (k ,r,s-1 ) i,r,r+1,r+1 ,w0 v 0   = - n-2-r =1 (c -c ) S k ,r,s ,w0 v 0 , and
Z n-2-r • • • • • Z 1 •   i∈I \I -1 ε(i)S (k ,r,s-1 ) i,r+1,n-+1,r+1 ,w0 v 0   = c -1 h=1 (c h -c h ) n-2-r h= +1 c h S k ,r,s-1 ,w0 v 0
for each 1 ≤ ≤ n -2 -r. Therefore by replacing (r, s) in Corollary 3.4.105 by (r + 1, s -1), we can deduce that

Z n-2-r • • • • • Z 1 • X + r+1 • X - r+1 • S k ,r,s-1 ,w0 v 0 = -([a 0 -a n-1 ] 1 + n -1 -s) 2 n-2-r =1 (c -c ) S k ,r,s ,w0 v 0 + CS k ,r,s-1 ,w0 v 0 = -([a 0 -a n-1 ] 1 + n -1 -s) 2 c n-1-r S k ,r,s ,w0 v 0 + CS k ,r,s-1 ,w0 v 0
for certain constant C ∈ F p . Note that we use the identity (3.4.108) for the last equality . By our assumption, we know that S k ,r,s-1 ,w0 v 0 ∈ V 0 . Hence we can deduce

S k ,r,s ,w0 v 0 ∈ V 0 since ([a 0 -a n-1 ] 1 + n -1 -s) 2 c n-1-r = 0.
Corollary 3.4.125. We have S k ,w0 v 0 ∈ V 0 .

Proof. By (3.4.103) and Proposition 3.4.124 we deduce that

S k ,r v 0 ∈ V 0 ⇒ S k ,r-1 v 0 ∈ V 0
for each 2 ≤ r ≤ n -1. Then by (3.4.102) and the definition of V 0 , we finish the proof.

We write β for n-1 r=1 α r to lighten the notation. Lemma 3.4.126. Given a Jacobi sum S k,w0 , we have

X β,k1,n • S k,w0 = (-1) k1,n+1 S k ,w0
where k = (k i,j ) satisfies k 1,n = 0 and k i,j = k i,j otherwise.

Proof. This is a special case of Lemma 3.4.11 when α 0 = β and m = k 1,n .

From now on, whenever we want to view the notation µ w0 0 as a weight, namely to fix a lift of µ w0 0 ∈ X(T )/(p -1)X(T ) into X reg 1 (T ), we always mean µ w0 0 = (a 0 + p -1, a n-2 , • • • , a 1 , a n-1 -p + 1) ∈ X(T ). In particular, we have

(1, n) • µ w0 0 + pβ = µ * . We recall the operators X alg β,k from the beginning of Section 3.4.

such that the following equality in Grothendieck group holds

i>0 V i (µ w0 0 ) = F (µ * ).
This equality implies that V 1 (µ w0 0 ) = F (µ * ) and V i (µ w0 0 ) = 0 for all i ≥ 2. By [Jan03] II.8.19 (2) we also know that

V (µ w0 0 )/V 1 (µ w0 0 ) ∼ = F (µ w0 0 ),
and thus we have shown that V (µ w0 0 ) = F (µ w0 0 ) + F (µ * ) in the Grothendieck group.

Proposition 3.4.133. We have V = H 0 (µ w0 0 ).

Proof. By Lemma 3.4.132, we have the natural surjection

H 0 (µ w0 0 ) F (µ * )
which induces a morphism H 0 (µ w0 0 ) µ * → F (µ * ) µ * . Now we consider H 0 (µ w0 0 ) as a L 1 -representation where L 1 ∼ = G m × GL n-1 is the standard Levi subgroup of G which contains U 1 as a maximal unipotent subgroup. For any λ ∈ X L1 (T ) + (c.f. (3.5.1)) we use the notation H 0 L1 (λ) for the L 1 -dual Weyl module defined at the beginning of Section 3.4. The dual Weyl module H 0 (µ w0 0 ) is the mod p reduction of a lattice V Zp in the unique irreducible algebraic representation V Qp of G such that V U Qp µ w 0 0 = 0. As the category of finite dimensional algebraic representations of L 1 in characteristic 0 is semisimple, V decomposes into a direct sum of characteristic 0 irreducible representations of L 1 . More precisely, we have the decomposition

V Qp | L1 = λ∈X L 1 (T )+ (V Qp ) U 1 λ =0 m λ V L1 (λ)
where V L1 (λ) is the unique (up to isomorphism) irreducible algebraic representation of L 1 such that V L1 (λ) U1 λ = 0 and

m λ := dim Qp V U1 Qp λ .
Therefore in the Grothendieck group of algebraic representations of L 1 over F p , we have

[H 0 (µ w0 0 )]| L1 = λ∈X L 1 (T )+ H 0 (µ w 0 0 ) U 1 λ =0 m λ [H 0 L1 (λ)] (3.4.134)
as by Corollary 3.4.80 H 0 (µ w0 0 ) U 1 is the mod p reduction of V U1 Zp and

V U1 Zp ⊗ Zp Q p = V U1 Qp . We say that µ * ↑ L1 λ 116CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
if there exists w ∈ W L1 (see the beginning of Section 3.5) such that λ = w • µ * and µ * ≤ λ.

Assume that there exists a λ ∈ X L1 (T ) + such that µ * ↑ L1 λ and that H 0 (µ w0 0 ) U 1 λ = 0. We denote by v alg {n-1},m the vector in H 0 (µ w0 0 ) U 1 λ = 0 given by Corollary 3.4.80. We note that by Corollary 3.4.80 the vector in H 0 (µ w0 0 ) U 1 µ * is v alg {n-1},m (see (3.4.130)). As µ * ↑ L1 λ, we must firstly have

n-1 i=1 m i = [a 0 -a n-1 ] 1 + n -2.
By the last statement in Corollary 3.4.80, we have

λ = a 0 + p -1 - n-1 i=1 m i , a n-2 + m 1 , • • • , a 1 + m n-2 , a n-1 -p + 1 + m n-1 = (a n-1 -n + 2, a n-2 + m 1 , • • • , a 1 + m n-2 , a n-1 -p + 1 + m n-1 ). (3.4.135) Recall η = (n -1, n -2, • • • , 1, 0)
. We notice that µ * -η lies in the lowest restricted L 1 -alcove in the sense that 0 < µ * , α ∨ < p for all α ∈ Φ +

L1

(3.4.136)

where Φ + L1 is the positive roots of L 1 defined at the beginning of Section 3.5. As we assume that (a n-1 , • • • , a 0 ) is n-generic, it is easy to see the following

   a n-2 + m 1 -(a n-1 -p + 1 + m n-1 ) ≤ p + 1 + a n-2 -a n-1 + m 1 < 2p; a n-2 + m 1 -(a 1 + m n-2 ) ≤ a n-2 + m 1 -a 1 ≤ [a 0 -a 1 ] 1 < p; a n-3 + m 2 -(a n-1 -p + 1 + m n-1 ) ≤ [a n-3 -a n-1 ] 1 + m 2 ≤ [a n-2 -a n-1 ] 1 < p,
so that we know that λ -η lies in either the lowest L 1 -alcove in the sense of (3.4.136) (if we replace µ * by λ) or the p-restricted L 1 -alcove described by the conditions

               p < λ, n-1 i=2 α i ∨ < 2p 0 < λ, n-2 i=2 α i ∨ < p 0 < λ, n-1 i=3 α i ∨ < p and 0 < λ, α ∨ < p for all α ∈ ∆ L1
where ∆ L1 := {α i | 2 ≤ i ≤ n -1} is the positive simple roots in Φ + L1 . In the first case, if λ -η lies in the lowest L 1 -alcove, as we assume that µ * ↑ L λ, we must have λ = µ * ; in the second case, we must have

λ = (2, n) • µ * + p n-1 i=2 α i = (a n-1 -n + 2, a 0 + p, a n-3 , • • • , a 1 , a n-2 + n -2 -p) which means by (3.4.135) that m = (m 1 , • • • , m n-1 ) = ([a 0 -a n-2 ] 1 + 1, 0, • • • , 0, a n-2 -a n-1 + n -3).
This implies a n-2 -a n-1 + n -1 = m n-1 ≥ 0, which is a contradiction to the n-generic assumption on (a n-1 , • • • , a 0 ). Therefore we must have λ = µ * . Hence we deduce by (3.4.134) and the strong linkage principle [Jan03] II.2.12 (1) that F L1 (µ * ) (see the beginning of Section 3.5 for notation) has multiplicity one in JH L1 (H 0 (µ w0 0 )| L1 ) and is actually a direct summand.

On the other hand, as F L1 (µ * ) is obviously an L 1 -subrepresentation of F (µ * ), we know that the surjection of G-representation H 0 (µ w0 0 ) F (µ * ) induces an isomorphism of L 1 -representation on the direct summand F L1 (µ * ) on both sides with multiplicity one, by restriction from G to L 1 . In particular, we know that the map H 0 (µ w0 0 ) U 1 µ * → F (µ * ) µ * is a bijection, and therefore the composition

V → H 0 (µ w0 0 ) F (µ * ) is non-zero as H 0 (µ w0 0 ) U 1 µ * = F p [v alg {n-1},m ] = F p [S k ,w0
v 0 ] by Lemma 3.4.129. Hence, we have a surjection

V F (µ * ).
Combining this surjection with the injection

V → H 0 (µ w0 0 ),
we finish the proof by Lemma 3.4.132.

Theorem 3.4.137. Assume that (a n-1 , • • • , a 0 ) is n-generic in the lowest alcove (c.f. Definition 3.4.5). Then H 0 (µ w0 0 ) ⊆ V 0 . In particular, we have

F (µ * ) ∈ JH(V 0 ).
Proof. The first inclusion is a direct consequence of Proposition 3.4.133 together with Corollary 3.4.125. The second inclusion follows from the first as we have F (µ * ) ∈ JH(H 0 (µ w0 0 )).

Before we end this section, we need several remarks to summarize the proof, and to clarify the necessity for all the constructions. Remark 3.4.138. If we assume that for all

2 ≤ k ≤ n -2 [a 0 -a n-1 ] 1 + n -2 < a k -a k-1 , (3.4.139)
then we can actually show that S k 0 ,w0 v 0 ∈ H 0 (µ w0 0 )

[U ,U ] µ * using Corollary 3.4.29 and the case s = n -1 of Proposition 3.4.78, and thus

V 0 = H 0 (µ w0 0 ).
Moreover, under the condition (3.4.139), we can even prove that the set

{S k,w0 v 0 | k ∈ Λ w0 }
forms a basis for H 0 (µ w0 0 ) µ * . On the other hand, if we have

[a 0 -a n-1 ] 1 + n -2 ≥ a k -a k-1
for some 2 ≤ k ≤ n -2, then we can show that

F (µ s k w0 0 ) ∈ JH(V 0 ) 118CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
which means that the inclusion H 0 (µ w0 0 ) ⊆ V 0 is actually strict.

In fact, through the proof of Proposition 3.4.124, the subrepresentation of π 0 generated by S k ,r,s v 0 is shrinking if r is fixed and s is growing. Therefore the subrepresentation of π 0 generated by S k ,r v 0 shrinks as r decreases. Finally, we succeeded in shrinking from V 0 to V which can be identified with H 0 (µ w0 0 ).

Remark 3.4.140. We need to emphasize that the choice of the operators X + r and X - r for 2 ≤ r ≤ n-1 are crucial. For example, the operator

t∈Fp t p-2 w 0 u αr (t)w 0 ∈ F p [G(F p )]
for some 2 ≤ r ≤ n -2 does not work in general. The reason is that, as one can check by explicit computation, applying such operator to S kw0 v 0 for some k ∈ Λ w0 will generally give us a huge linear combination of Jacobi sum operators. From our point of view, it is basically impossible to compute such a huge linear combination explicitly and systematically. Instead, as stated in Proposition 3.4.101, our operators X + r and X - r can be computed systematically, even though the computation is still complicated.

The motivation of the choice of operators X + r and X - r can be roughly explained as follows. First of all, we need one 'weight raising operator' X + and one 'weight lowering operator' X -. These are two operators lying in a subalgebra

F p X + , X -of F p [G(F p )] such that F p X + , X -∼ = F p [GL 2 (F p )].
We start with the vector S k,w0 v 0 for some k ∈ Λ w0 . We apply the operator X -once and then X + once, the result is a vector with the same T (F p )-eigencharacter µ * . We observe that S k,w0 v 0 is in general not an eigenvector of the operator X + • X -because the representation π 0 , after restricting from F p [G(F p )] to F p X + , X -, is highly non-semisimple. The naive expectation is that we just take the difference

X + • X -• S k,w0 v 0 -cS k,w0 v 0
for some constant c ∈ F p , and then repeat the procedure by applying some other operators similar to X + and X -. The case n = 3 is easy. In the case n = 4, the operator

t∈Fp t p-2 w 0 u α2 (t)w 0 ∈ F p [GL 4 (F p )]
is not well behaved as we explained in this remark, and therefore we are forced to use our X - 2 to replace t∈Fp t p-2 w 0 u α2 (t)w 0 . Now we consider the general case, and it is possible for us to carry on an induction step. We have a sequence of growing subgroups of G

P {n-1} P {n-2,n-1} • • • P {2,••• ,n-1} and L {n-1} L {n-2,n-1} • • • L {2,••• ,n-1}
where P {r,••• ,n-1} is the standard parabolic subgroup corresponding to the simple roots α k for r ≤ k ≤ n -1 and L {r,••• ,n-1} is its standard Levi subgroup. Technically speaking, constructing the vector S k ,r+1 ,w0 v 0 (for some 1 ≤ r ≤ n -2) from S k 0 ,w0 v 0 should be reduced to Corollary 3.4.125 when we replace G by its Levi subgroup L {r+1,••• ,n-1} . In other words, to construct S k ,r+1 ,w0 v 0 from S k 0 ,w0 v 0 we only need the operators

X + k , X - k ∈ F p [L {r+2,••• ,n-1} (F p )] F p [L {r+1,••• ,n-1} (F p )]
for all r + 2 ≤ k ≤ n -1.

In order to construct S k ,r ,w0 v 0 from S k ,r+1 ,w0 v 0 , we only need to prove Proposition 3.4.124. Then we summarize the proof of Proposition 3.4.124 as the following: for some a ∈ F × p and b ∈ F p X + r+1 • X - r+1 • S k ,r,s-1 ,w0 v 0 ≡ aS k ,r,s ,w0 v 0 + bS k ,r,s-1 ,w0 v 0 + error terms and the error terms can be killed by combinations of the operators X + k , X - k for r + 2 ≤ k ≤ n -1.

Mod p local-global compatibility

In this section, we state and prove our main results on mod p local-global compatibility, which is a global application of our local results of Sections 3.3 and 3.4. In the first two sections, we recall some necessary known results on algebraic automorphic forms and Serre weights, for which we closely follow [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], and [BLGG]. We first fix some notation for the whole section. Let P ⊇ B be an arbitrary standard parabolic subgroup and N its unipotent radical. We denote the opposite parabolic by P -:= w 0 P w 0 with corresponding unipotent radical N -:= w 0 N w 0 . We fix a standard choice of Levi subgroup L := P ∩ P -⊆ G. We denote the positive roots of L defined by the pair (B ∩ L, T ) by Φ + L . We use

X L (T ) + := {λ ∈ X(T ) | λ, α ∨ > 0 for all α ∈ Φ + L } (3.5.1)
to denote the set of dominant weights with respect to the pair (B ∩ L, T ). We denote the Weyl group of L by W L and identify it with a subgroup of W . The longest Weyl element in W L is denoted by w L 0 . We define the affine Weyl group W L of L as the semi-direct product of W L and X(T ) with respect to the natural action of W L on X(T ). Therefore W L has a natural embedding into W . We define the subgroups P , L, • • • of G in the obvious similar fashion.

We also need to define several open compact subgroups of L(Q p ). We define

K L := L(Z p ),
and via the mod p reduction map

red L : K L = L(Z p ) L(F p )
we also define K L (1), I L (1), and I L as follows:

K L (1) := (red L ) -1 (1) ⊆ I L (1) := (red L ) -1 (U (F p ) ∩ L(F p )) ⊆ I L := (red L ) -1 (B(F p ) ∩ L(F p )). (3.5.2)
For any dominant weight λ ∈ X(T ) + , we let

H 0 L (λ) := Ind L B∩L w L 0 λ alg /Fp
be the associated dual Weyl module of L. We also write F L (λ) := soc L H 0 L (λ) for its irreducible socle as an algebraic representation of L. Through a similar argument presented at the beginning of Section 3.4, the notation F L (λ) is well defined as an irreducible representation of L(F p ) if λ ∈ T (F p ) is p-regular, namely lies in the image of X reg 1 (T ) → X(T )/(p -1)X(T ). We will sometimes abuse the notation F L (λ) for F L (λ) ⊗ Fp F or F L (λ) for F L (λ) ⊗ Fp F p in the literature. We will emphasize the abuse of the notation F L (λ) each time we do so. We introduce some specific standard parabolic subgroups of G. Fix integers i 0 and j 0 such that 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1, and let i 1 and j 1 be the integers determined by the equation

i 0 + i 1 = j 0 + j 1 = n -1.
(3.5.3)

We let P i1,j1 ⊃ B be the standard parabolic subgroup of G = GL n corresponding to the subset {α k | j 0 + 1 ≤ k ≤ i 0 } of ∆. By specifying the notation for general P to P i1,j1 , we can define P - i1,j1 , L i1,j1 , N i1,j1 and N - i1,j1 . We can naturally embeds GL j1-i1+1 into G with its image denoted by G i1,j1 such that L i1,j1 = G i1,j1 T :

GL j1-i1+1 ∼ → G i1,j1 → L i1,j1 → P i1,j1 → G.
(3.5.4)

We define T i1,j1 to be the maximal tori of G i1,j1 that is contained in T , and define X(T i1,j1 ) to be the character group of T i1,j1 . If i 1 and j 1 are clear from the context (or equivalently i 0 and j 0 are clear) then we often write P , P -L, N , and N -for P i1,j1 , P - i1,j1 , L i1,j1 , N i1,j1 , and N - i1,j1 , respectively.

The space of algebraic automorphic forms

Let F/Q be a CM field with maximal totally real subfield F + . We write c for the generator of Gal(F/F + ), and let S + p (resp. S p ) be the set of places of F + (resp. F ) above p. For v (resp. w) a finite place of F + (resp. F ) we write k v (resp. k w ) for the residue field of F + v (resp. F w ). From now on, we assume that • F/F + is unramified at all finite places;

• p splits completely in F . Note that the first assumption above excludes F + = Q. We also note that the second assumption is not essential in this section, but it is harmless since we are only interested in G Qp -representations in this paper. Every place v of F + above p further decomposes and we often write v = ww c in F .

There exists a reductive group G n/F + satisfying the following properties (c.f. [BLGG], Section 2):

• G n is an outer form of GL n with G n/F ∼ = GL n/F ,
• G n is a quasi-split at any finite place of F + ;

• G n (F + v ) U n (R) for all v|∞.
By [START_REF] Clozel | Automorphy for some -adic lifts of automorphic mod representations[END_REF], Section 3.3, G n admits an integral model

G n over O F + such that G n × O F + O F + v is reductive if v is a finite place of F + which splits in F . If v is
such a place and w is a place of F above v, then we have an isomorphism

ι w : G n (O F + v ) ∼ → G n (O Fw ) ∼ → GL n (O Fw ).
(3.5.5)

We fix this isomorphism for each such place v of F + . Define

F + p := F + ⊗ Q Q p and O F + ,p := O F + ⊗ Z Z p . If W is an O E -module endowed with an action of G n (O F + ,p ) and U ⊂ G n (A ∞,p F + ) × G n (O F + ,p
) is a compact open subgroup, the space of algebraic automorphic forms on G n of level U and coefficients in W , which is also an O E -module, is defined as follows:

S(U, W ) := f : G n (F + )\G n (A ∞ F + ) → W | f (gu) = u -1 p f (g) ∀ g ∈ G n (A ∞ F + ), u ∈ U
with the usual notation u = u p u p for the elements in U .

MOD P LOCAL-GLOBAL COMPATIBILITY 121

We say that the level U is sufficiently small if

t -1 G n (F + )t ∩ U
has finite order prime to p for all t ∈ G n (A ∞ F + ). We say that U is unramified at a finite place v of

F + if it has a decomposition U = G n (O F + v )U v for some compact open U v ⊂ G n (A ∞,v F + ).
If w is a finite place of F , then we say, by abuse of notation, that w is an unramified place for U or U is unramified at w if U is unramified at w| F + .

For a compact open subgroup U of G n (A ∞,p F + ) × G n (O F + ,p ), we let P U denote the set consisting of finite places w of F such that

• w| F + is split in F , • w / ∈ S p ,
• U is unramified at w.

For a subset P ⊆ P U of finite complement and closed with respect to complex conjugation we write

T P = O E [T (i) w , w ∈ P, i ∈ {0, 1, • • • , n}]
for the universal Hecke algebra on P, where the Hecke operator T (i) w acts on S(U, W ) via the usual double coset operator

ι -1 w GL n (O Fw ) w Id i 0 0 Id n-i GL n (O Fw )
where w is a uniformizer of O Fw and Id i is the identity matrix of size i. The Hecke algebra T P naturally acts on S(U, W ).

Recall that we assume that p splits completely in F . Following [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 7.1 we consider the subset (Z n + )

Sp 0 consisting of dominant weights a = (a w ) w where a w = (a 1,w , a 2,w , • • • , a n,w ) satisfying a i,w + a n+1-i,w c = 0 (3.5.6) for all w ∈ S p and 1 ≤ i ≤ n. We let

W a w := M a w (O Fw ) ⊗ O Fw O E
where the M a w (O Fw ) is O Fw -specialization of the dual Weyl module associated to a w (c.f. [EGH13], Section 4.1.1); by condition (3.5.6), one deduces an isomorphism of

G n (O F + v )-representations W a w • ι w ∼ = W a w c • ι w c . Therefore, by letting W a v := W a w • ι w for any place w|v, the O E -representation of G n (O F + ,p ) W a := v|p W a v is well-defined.
For a weight a ∈ (Z n + ) Sp 0 , let us write S a (Q p ) to denote the inductive limit of the spaces

S(U, W a )⊗ O E Q p over the compact open subgroups U ⊂ G n (A ∞,p F + ) × G n (O F + ,p
). (Note that the transition maps are induced, in a natural way, from the inclusions between levels U .) Then S a (Q p ) has a natural left action of G n (A ∞ F + ) induced by right translation of functions. We briefly recall the relation between the space A of classical automorphic forms and the previous spaces of algebraic automorphic forms in the particular case which is relevant to us. Fix an isomorphism ı : Q p

p ∼ → C induces an isomorphism of smooth G n (A ∞ F + )-representations S a (Q p ) ⊗ Q p ,ı C ı -→ Hom Gn(F + ⊗ Q R) (σ ∨ a , A)
for any a ∈ (Z n + )

Sp 0

The following theorem guarantees the existence of Galois representations attached to automorphic forms on the unitary group G n . We let | |

G n (A ∞ F + )-subrepresentation of S a (Q p ).
Then there exists a continuous semisimple representation

r Π : G F → GL n (Q p )
such that

(i) r c Π ⊗ ε n-1 ∼ = r ∨ Π ;
(ii) for each place w above p, the representation r Π | G Fw is de Rham with Hodge-Tate weights

HT(r Π | G Fw ) = {a 1,w + (n -1), a 2,w + (n -2), • • • , a n,w }; (iii) if w|p is a place of F and v := w| F + splits in F , then WD(r Π | G Fw ) F-ss ∼ = rec w ((Π v • ι -1 w ) ⊗ | • | 1-n 2 ).
We note that the fact that (iii) holds without semi-simplification on the automorphic side is one of the main results of [?]. We also note that property (iii) says that the restriction to G Fw is compatible with the local Langlands correspondence at w, which is denoted by rec w .

Serre weights and potentially crystalline lifts

In this section, we recall the relation of Serre weights and potentially crystalline lifts via (inertial) local Langlands correspondence. Definition 3.5.9. A Serre weight for G n is an isomorphism class of an absolutely irreducible smooth F p -representation V of G n (O F + ,p ). If v is a place of F + above p, then a Serre weight at v is an isomorphism class of an absolutely irreducible

F p -smooth representation V v of G n (O F + v ).
Finally, if w is a place of F above p, a Serre weight at w is an isomorphism class of an absolutely irreducible F p -smooth representation V w of GL n (O Fw ).

We will often say a Serre weight for a Serre weight for G n if G n is clear from the context. Note that if V v is a Serre weight at v, there is an associated Serre weight at w|v defined by V v • ι -1 w . As explained in [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 7.3, a Serre weight V admits an explicit description in terms of GL n (k w )-representations. More precisely, let w be a place of F above p and write v := w| F + . For any n-tuple of integers a w := (a 1,w , a 2,w , • • • , a n,w ) ∈ Z n + , that is restricted (i.e., 0 ≤ a i,w -a i+1,w ≤ p -1 for i = 1, 2, • • • , n -1), we consider the Serre weight F (a w ) := F (a 1,w , a 2,w , • • • , a n,w ), as defined in [START_REF] Emerton | Weight cycling and Serre-type conjectures for unitary groups[END_REF], Section 4.1.2. It is an irreducible F p -representation of GL n (k w ) and of G n (k v ) via the isomorphism ι w . Note that F (a 1,w , a 2,w ,

• • • , a n,w ) ∨ • ι w c ∼ = F (a 1,w , a 2,w , • • • , a n,w ) • ι w as G n (k v )- representations, i.e. F (a w c ) • ι w c ∼ = F (a w ) • ι w if a i,w + a n+1-i,w c = 0 for all 1 ≤ i ≤ n. Hence, if a = (a w ) w ∈ (Z n + )
Sp 0 that is restricted, then we can set F a v := F (a w ) • ι w for w|v. We also set

F a := v|p F a v 3.
∈ (Z n + ) Sp 0 such that V has a decomposition V ∼ = v|p V v where the V v are Serre weights at v satisfying V v • ι -1 w ∼ = F (a w
). Recall that we write F for the residue field of E. Definition 3.5.10. Let r : G F → GL n (F) be an absolutely irreducible continuous Galois representation and let V be a Serre weight for G n . We say that r is automorphic of weight V (or that V is a Serre weight of r) if there exists a compact open subgroup U in G n (A ∞,p F ) × G n (O F + ,p ) unramified above p and a cofinite subset P ⊆ P U such that r is unramified at each place of P and S(U, V ) mr = 0 where m r is the kernel of the system of Hecke eigenvalues α : T P → F associated to r, i.e.

det (1 -r ∨ (Frob w )X) = n j=0 (-1) j (N F/Q (w)) ( j 2 ) α(T (j) w )X j
for all w ∈ P.

We write W (r) for the set of automorphic Serre weights of r. Let w be a place of F above p and v = w| F + p . We also write W w (r) for the set of Serre weights F (a w ) such that

(F (a w ) • ι w ) ⊗   v ∈S + p \{v} V v   ∈ W (r)
where

V v are Serre weights of G n (O F + v
) for all v ∈ S + p \{v}. We often write W (r| G Fw ) and W w (r| G Fw ) for W (r) and W w (r) respectively, when the given r| G Fw is clearly a restriction of an automorphic representation r to G Fw .

Fix a place w of F above p and let v = w| F + p . We also fix a compact open subgroup U of G n (A ∞,p F ) × G n (O F + ,p ) which is sufficiently small and unramified above p. We may write

U = G n (O F + v ) × U v . If W is an O E -module with an action of v ∈S + p \{v} G n (O F + v
), we define

S(U v , W ) := lim -→ Uv S(U v • U v , W )
where the limit runs over all compact open subgroups U v of G n (O F + v ), endowing W with a trivial G n (O F + v )-action. Note that S(U v , W ) has a smooth action of G n (F + v ) (given by right translation) and hence of GL n (F w ) via ι w . We also note that S(U v , W ) has an action of T P commuting with the smooth action of G n (F + v ), where P is a cofinite subset of P U .

Lemma 3.5.11

([EGH13], Lemma 7.4.3). Let U be a compact open subgroup of G n (A ∞,p F )×G n (O F + ,p
) which is sufficiently small and unramified above p, and P a cofinite subset of P U . Fix a place w of F above p and let v = w| F + p . Let V ∼ = v ∈S + p V v be a Serre weight for G n . Then there is a natural isomorphism of T P -modules

Hom Gn(O F + v ) (V ∨ v , S(U v , V )) ∼ → S(U, V )
where V := v ∈S + p \{v} V v . We now recall some formalism related to Deligne-Lusztig representations from Section 3.4.3. Let w be a place of F above p. For a positive integer m, let k w,m /k w be an extension satisfying [k w,m :

k w ] = m, and let T be a F -stable maximal torus in GL n/k w where F is the Frobenius morphism. We have an identification from [START_REF] Herzig | The weight in a Serre-type conjecture for tame n-dimensional Galois representations[END_REF], Lemma 4.7

T(k w ) ∼ -→ j k × w,nj
where n ≥ n j > 0 and j n j = n; the isomorphism is unique up to j Gal(k w,nj /k w )-conjugacy. In particular, any character θ : T(k w ) → Q × p can be written as θ = ⊗ j θ j where θ j : k × w,nj → Q × p . Given a F -stable maximal torus T and a primitive character θ, we consider the Deligne-Lusztig representation R θ T of GL n (k w ) over Q p defined in Section 3.4.3. Recall from Section 3.4.3 that Θ(θ j ) is cuspidal representation of GL nj (k w ) associated to the primitive character θ j , we have

R θ T ∼ = (-1) n-r • Ind GLn(kw) Pn(kw) (⊗ j Θ(θ j ))
where P n is the standard parabolic subgroup containing the Levi j GL nj and r denotes the number of its Levi factors.

Let F w,m := W (k w,m )[ 1 p ] for a positive integer m. We consider θ j as a character on O ×

σ θ j • Art -1 Fw,n j  
where θ j is a primitive character on k × w,nj of niveau n j for each j = 1, • • • , r. Recall that Art Fw,n j :

F × w,nj → W ab Fw,n j
is the isomorphism of local class field theory, normalized by sending the uniformizers to the geometric Frobenius.

We quickly review inertial local Langlands correspondence. Moreover, if τ ∼ = ⊕ r j=1 τ j and the τ j are pairwise distinct, then σ(τ ) ∼ = R θ T and τ ∼ = rec(θ) for a maximal torus T in GL n/F p and a primitive character θ :

T(F p ) → Q × p .
The following theorem provides a connection between Serre weights and potentially crystalline lifts, which will be useful for the main result, Theorem 3.5.44. Theorem 3.5.13 ([LLL16], Proposition 4.2.5). Let w be a place of F above p, T a maximal torus in GL n/k w , θ = r j=1 θ j : T(k w ) → Q × p a primitive character such that θ j are pairwise distinct, and V w a Serre weight at w for a Galois representation r : G F → GL n (F).

Assume that V w is a Jordan-Hölder constituent in the mod p reduction of the Deligne-Lusztig representation R θ T of GL n (k w ). Then r| G Fw has a potentially crystalline lift with Hodge-Tate weights {-(n -1), -(n -2), • • • , 0} and Galois type rec(θ).

For a given automorphic Galois representation r : G F → GL n (F), it is quite difficult to determine if a given Serre weight is a Serre weight of r. Thanks to the work of [BLGG], we have the following theorem, in which we refer the reader to [BLGG] for the unfamiliar terminology.

Theorem 3.5.14 ([BLGG], Theorem 4.1.9). Assume that if n is even then so is n[F + :Q] 2 , that ζ p ∈ F , and that r : G F → GL n (F) is an absolutely irreducible representation with split ramification. Assume further that there is a RACSDC automorphic representation Π of GL n (A F ) such that We now consider characteristic 0 lifts of the mod p Jacobi sum operators above.

S i1,j1 :=    A∈U w L 0 (Fp)   n-i1-1 =n-j1 A , +1 k i 1 ,j 1 , +1   A    w L 0 and S i1,j1, :=    A∈U w L 0 (Fp)   n-i1-1 =n-j1 A , +1 k i 1 ,j 1 , , +1   A    w L 0 .
We also let

S i1,j1 0 :=    A∈U w L 0 (Fp)   n-i1-1 =n-j1 A , +1 k i 1 ,j 1 ,0 , +1   A    w L 0
where k i1,j1,0 = (k i1,j1,0 i,j

) i,j ∈ {0, • • • , p -1} |Φ + w L 0 | satisfies k i1,j1,0 i,j := i 1 -j 1 + 1 + [b i1 -b j1 ] 1 if n -j 1 ≤ i = j -1 ≤ n -i 1 -1; 0 if j ≥ i + 2. (3.5.24)
Note that S i1,j1 , S i1,j1, , S i1,j1 0 are Teichmüler lifts of S i1,j1 , S i1,j1, , S k i 1 ,j 1 ,0 ,w L 0 , respectively. We will also consider the Teichmüler lifts of S i1,j1 1 and S i1,j1, 1 as follows:

S i1,j1 1 :=    A∈U w i 1 ,j 1 1 (Fp) A    w i1,j1 1 and S i1,j1, 1 :=    A∈U w i 1 ,j 1 , 1 (Fp) A    w i1,j1, 1 .
We recall the operator Ξ n ∈ G(Q p ) from (3.4.49). Note that µ i1,j1

1 : T (F p ) → O × E is the Teichmüler lift of µ i1,j1
1 . We also recall κ

, • • • , a 0 ). We define κ (1) i1,j1 , κ (2) i1,j1 , κ i1,j1 ∈ Z × p , ε i1,j1 = ±1 and P i1,j1 ∈ Z × p by replacing n and (a n-1 , • • • , a 1 , a 0 ) by j 1 -i 1 + 1 and (b j1 + j 1 -i 1 -1, b j1-1 , • • • , b i1+1 , b i1 -j 1 + i 1 + 1
) respectively with b k as at the beginning of Section 3.5.3. Proposition 3.5.25. Assume that µ ,i1,j1 is 2n-generic. Let

Π i1,j1 := Ind G(Qp) B(Qp) χ i1,j1
be a tamely ramified principal series where χ

i1,j1 = χ i1,j1 n-1 ⊗ • • • ⊗ χ i1,j1 0 : T (Q p ) → E × is a smooth character satisfying χ | T (Zp) ∼ = µ i1,j1
1 . Then we have the identity

S i1,j1, • S i1,j1, 1 • (Ξ n ) j1-i1-1 = p (j1-i1-1)(i1+1) κ i1,j1   n-1 k=n-j1+i1+1 χ i1,j1 k (p)   S i1,j1 • S i1,j1
1 on the 1-dimensional space (Π i1,j1 ) I(1), µ i 1 ,j 1 1 . Proof. By Lemma 3.4.54 we know that

(Ξ n ) j1-i1-1 • U j1-i1-1 n = S (w * ) j 1 -i 1 -1 . 130CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE
Then by Lemma 3.4.51 and the fact (w i1,j1, 1 ) + ((w * ) j1-i1-1 ) = (w i1,j1, 1 (w * ) j1-i1-1 (w i1,j1 1 ) -1 ) + (w i1,j1 1 ) + 2(j 1 -i 1 -1)i 1 we deduce that

S w i 1 ,j 1 , 1 • S (w * ) j 1 -i 1 -1 = p (j1-i1-1)i1 S w i 1 ,j 1 , 1 (w * ) j 1 -i 1 -1 (w i 1 ,j 1 1 ) -1 • S w i 1 ,j 1 1 .
Therefore it remains to show that

S i1,j1, • S w i 1 ,j 1 , 1 (w * ) j 1 -i 1 -1 (w i 1 ,j 1 1 ) -1 = p j1-i1-1 κ i1,j1 S i1,j1 on the 1-dimensional space (Π i1,j1 ) I(1), µ i 1 ,j 1 = S w i 1 ,j 1 1 (Π i1,j1 ) I(1), µ i 1 ,j 1 1 .
We observe by Lemma 3.4.51 that

S w i 1 ,j 1 , • S w i 1 ,j 1 , 1 (w * ) j 1 -i 1 -1 (w i 1 ,j 1 1 ) -1 = p j1-i1-1 S w i 1 ,j 1
and therefore by composing S i1,j1 0 it remains to show that

S i1,j1 0 • S w i 1 ,j 1 , = p j1-i1-1 (κ (2) i1,j1 ) -1 S i1,j1,
(3.5.26) on (Π i1,j1 ) I(1), µ i 1 ,j 1 , and S i1,j1

0 • S w i 1 ,j 1 = p j1-i1-1 (κ (1)
i1,j1 ) -1 S i1,j1 (3.5.27) on (Π i1,j1 ) I(1), µ i 1 ,j 1 . But these can be checked by the same argument as in Corollary 3.4.66.

We state here a generalization of the Theorem 3.4.36. Recall the definition of π i1,j1 * from (3.5.15).

Theorem 3.5.28. The constituent F (µ ) has multiplicity one in π i1,j1 * .

Proof. This is Corollary 3.4.47 if we replace µ i1,j1 π by µ .

We define a characteristic zero principal series

( π i1,j1 * ) • := Ind G(Fp) B(Fp) ( µ ,i1,j1 ) w0 which is an O E -lattice in ( π i1,j1 * ) • ⊗ O E E. Lemma 3.5.29. (i) For µ ∈ {µ i1,j1 , µ i1,j1, , µ i1,j1 1 , µ i1,j1, 1 }, we have dim Fp (π i1,j1 * ) U (Fp),µ = 1.
(ii) We have the following non-vanishing results:

S i1,j1 (π i1,j1 * ) U (Fp),µ i 1 ,j 1 = S i1,j1, (π i1,j1 * ) U (Fp),µ i 1 ,j 1 , = 0.
(iii) We also have the following non-vanishing results:

S i1,j1 1 (π i1,j1 * ) U (Fp),µ i 1 ,j 1 1 = (π i1,j1 * ) U (Fp),µ i 1 ,j 1 and S i1,j1 1 (π i1,j1 * ) U (Fp),µ i 1 ,j 1 , 1 = (π i1,j1
Proof. The statement (i) is immediate by Bruhat decomposition (3.4.4). Now we prove (ii). According to Lemma 3.4.51, (3.5.26) and (3.5.27) and Lemma 3.4.65, we deduce by mod p reduction with respect to the lattice ( π i1,j1 *

) • that S i1,j1 (π i1,j1 * ) U (Fp),µ i 1 ,j 1 = S i1,j1, (π i1,j1 * ) U (Fp),µ i 1 ,j 1 , = S k i 1 ,j 1 ,0 ,w L 0 (π i1,j1 * ) U (Fp),(µ ,i 1 ,j 1 ) w L 0 .
If we abuse the notation k i1,j1,0 for the tuple in {0,

• • • , p -1} |Φ + w 0 | satisfying k i1,j1,0 α = 0 for all α / ∈ Φ + w L 0
then by mod p reduction of first possibility of Proposition 3.4.60 we deduce that

S k i 1 ,j 1 ,0 ,w L 0 • S 0,w L 0 w0 = S k i 1 ,j 1 ,0 ,w0
on the 1-dimensional subspace (π i1,j1 *

) U (Fp),(µ ,i 1 ,j 1 ) w 0 . Thus we finish the proof of (ii) by

S k i 1 ,j 1 ,0 ,w0 (π i1,j1 *
) U (Fp),(µ ,i 1 ,j 1 ) w 0 = 0 which follows from Proposition 3.4.18. Finally we prove (iii). We only prove the first equality in (iii) as the same proof works for the second equality. By Lemma 3.4.20 we know that S 0,(w i 1 ,j 1 ) -1 w L 0 w0 (π i1,j1 *

) U (Fp),(µ ,i 1 ,j 1 ) w 0 = (π i1,j1 *

) U (Fp),µ i 1 ,j 1 and S 0,(w i 1 ,j 1 w i 1 ,j 1 1

) -1 w L 0 w0 (π i1,j1 *

) U (Fp),(µ ,i 1 ,j 1 ) w 0 = (π i1,j1 *

) U (Fp),µ i 1 ,j 1 1 .

Therefore it remains to show that S i1,j1 1

• S 0,(w i 1 ,j 1 w i 1 ,j 1 1

) -1 w L 0 w0 = S 0,(w i 1 ,j 1 ) -1 w L 0 w0

on the 1-dimensional subspace (π i1,j1 * ) U (Fp),(µ ,i 1 ,j 1 ) w 0 , which follows from the mod p reduction of Lemma 3.4.51 and the fact that (w i1,j1 1 ) + ((w i1,j1 w i1,j1 1 ) -1 w L 0 w 0 ) = ((w i1,j1 ) -1 w L 0 w 0 ). This completes the proof.

We define V i1,j1 and V i1,j1, to be the subrepresentations of π i1,j1 * generated by S i1,j1 (π i1,j1 *

) U (Fp),µ i 1 ,j 1 and S i1,j1, (π i1,j1 *

) U (Fp),µ i 1 ,j 1 , respectively. Similarly, we define V i1,j1 0 as the subrepresentation of π i1,j1 * generated by

S k i 1 ,j 1 ,0 (π i1,j1 *
) U (Fp),(µ ,i 1 ,j 1 ) w 0 .

Lemma 3.5.30. We have

V i1,j1 = V i1,j1, = V i1,j1 0 (3.5.31) and F (µ ) ∈ JH(V i1,j1 0
).

(3.5.32)

Note that M i1,j1 is a free O E -module of finite rank as M is a admissible smooth representation of G(Q p ) which is E -torsion free. For any O E -algebra A, we write M i1,j1 A for M i1,j1 ⊗ O E A. We similarly define M A .

Let T i1,j1 be the O E -module that is the image of T P in End O E (M i1,j1 ). Then T i1,j1 is a local O E -algebras with the maximal ideal m r , where, by abuse of notation, we write m r ⊆ T for the image of m r of T P . As the level U is sufficiently small, by passing to a sufficiently large E as in the proof of Theorem 4.5.2 of [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], we may assume that T i1,j1 E ∼ = E r for some r > 0. For any O E -algebra

A we write T i1,j1 A for T i1,j1 ⊗ O E A. We have M i1,j1 E = p M i1,j1 E [p E ],
where the sum runs over the minimal primes p of T i1,j1 and p E := pT i1,j1 E . Note that for any such p T i1,j1 E /p E ∼ = E. By abuse of notation, we also write p (resp. p E ) for its inverse image in T P (resp. T P E ). We also note that for any such p we have a surjection

M [p] M F [m r ] as m r = p + E T P . Definition 3.5.43. A non-zero vector v i1,j1 ∈ M i1,j1
F is said to be primitive if there exists a vector v i1,j1 ∈ M i1,j1 [p] that lifts v i1,j1 , for certain minimal prime p of T.

Note that the G(Q p )-subrepresentation of M E generated by a lift v i1,j1 of a primitive element v i1,j1 is irreducible and actually lies in

M E [p E ].
Now we can state our main results in this paper. Recall that by ρ 0 we always mean an n-dimensional ordinary representation of G Qp as described in (3.3.1).

Theorem 3.5.44. Fix a pair of integers (i 0 , j 0 ) satisfying 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1, and let (i 1 , j 1 ) be a pair of integers such that i 0 + i 1 = j 0 + j 1 = n -1. We also let r : G F → GL n (F) be an irreducible automorphic representation with r| G Fw ∼ = ρ 0 . Assume that • µ ,i1,j1 is 2n-generic;

• ρ i0,j0 is Fontaine-Laffaille generic.

Assume further that

{F (µ ) ∨ } ⊆ W w (r) ∩ JH((π i1,j1 * ) ∨ ) ⊆ {F (µ ) ∨ , F (µ ,i1,j1 ) ∨ }. (3.5.45)
Then there exists a primitive vector in S(U v , V )[m r ] I(1),µ i 1 ,j 1 1 . Moreover, for each primitive vector

v i1,j1 ∈ S(U v , V )[m r ] I(1),µ i 1 ,j 1 1 we have S i1,j1 • S i1,j1 1 v i1,j1 = 0 and S i1,j1, • S i1,j1, • (Ξ n ) j1-i1-1 v i1,j1 = ε i1,j1 P i1,j1 (b n-1 , • • • , b 0 ) • FL i0,j0 n (r| G Fw ) • S i1,j1 • S i1,j1 1 v i1,j1
where

ε i1,j1 = j1-1 k=i1+1 (-1) bi 1 -b k -j1+i1+1 and P i1,j1 (b n-1 , • • • , b 0 ) = j1-1 k=i1+1 j1-i1-1 j=1 b k -b j1 -j b i1 -b k -j ∈ Z × p .
Remark 3.5.46. The right inclusion of (3.5.45) is just Conjecture 3.5.16, which is now a theorem of Bao V. Le Hung (c.f. Remark 3.1.13 and [LLMPQ]). We also give an evidence for the left inclusion of (3.5.45) in Proposition 3.5.17 under some assumption of Taylor-Wiles type. As a result, the condition (3.5.45) can be removed under some standard Taylor-Wiles conditions.

Remark 3.5.47. If M i1,j1 is free as T i1,j1 -module, then all vectors in S(U v , V )[m r ] I(1),µ i 1 ,j 1 1 are primitive. As a result, one needs such a freeness result to remove the "primitive" condition. Under a stronger generic condition (compared to our Fontaine-Laffaille generic), it is possible to use results from [LLMPQ] to improve (3.5.45) to an equality

W w (r) ∩ JH((π i1,j1 * ) ∨ ) = {F (µ ) ∨ }
in which case one is able to prove the freeness result mentioned above through the technique in Section 5 of [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] under some standard global assumption. It is also possible to prove a freeness result over some enlarged Hecke algebra as in Section 5 of [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF], at least if (i 1 , j 1 ) = (0, n -1).

Proof. We firstly point out that M i1,j1 = 0, as S(U, (F (µ

) ∨ • ι w ) ⊗ V ) mr = 0 and F (µ ) is a factor of Ind K I µ i1,j1 1 = Ind G(Fp) B(Fp) µ i1,j1 1
.

Picking an embedding E → Q p , as well as an isomorphism ι :

Q p ∼ -→ C, we see that M i1,j1 Q p ∼ = Π m(Π) • Π I(1), µ i 1 ,j 1 1 v ⊗ (Π ∞,v ) U v , (3.5.48)
where the sum runs over irreducible representations

Π ∼ = Π ∞ ⊗ Π v ⊗ Π ∞,v of G n (A F + ) over Q p such that Π ⊗ ι C is a cuspidal automorphic representation of multiplicity m(Π) ∈ Z >0 with Π ∞ ⊗ ι C
being determined by the algebraic representation ( V ) ∨ and with associated Galois representation r Π lifting r ∨ (c.f. Lemma 3.5.7). We write δ for the modulus character of B(Q p ):

δ :=| | n-1 ⊗ | | n-2 ⊗ • • • ⊗ | | ⊗1
where | | is the (unramfied) norm character sending p to p -1 . For any Π contributing to (3.5.48), we have

(i) Π v ∼ = Ind G(Qp) B(Qp) (ψ ⊗ δ) for some smooth character ψ = ψ n-1 ⊗ ψ n-2 ⊗ • • • ⊗ ψ 1 ⊗ ψ 0 of T (Q p ) such that ψ| T (Zp) = µ i1,j1 1 | T (Zp)
, where ψ k are the smooth characters of Q × p .

(ii) r ∨ Π | G Fw is a potentially crystalline lift of r with Hodge-Tate weights {-

(n-1), -(n-2), • • • , -1, 0} and WD(r ∨ Π | G Fw ) F-ss ∼ = ⊕ n-1 k=0 ψ -1 k .
Here, part (i) follows from [EGH13], Propositions 2.4.1 and 7.4.4, and part (ii) follows from classical local-global compatibility (c.f. Theorem 3.5.8). Moreover, by Corollary 3.3.46, we have

FL i0,j0 n (ρ 0 ) = i0-1 k=j0+1 ψ i1+1+k (p) p (i 0 +j 0 )(i 0 -j 0 -1) 2
.

(3.5.49) (Note that we may identify ψ i1+1+k with Ω -1 k for j 0 < k < i 0 , where Ω k is defined in Corollary 3.3.46.) Now we pick an arbitrary primitive vector v i1,j1 ∈ M i1,j1

F [m r ] with a lift v i1,j1 ∈ M i1,j1 [p]. We set τ E := K v i1,j1 E ⊆ M E [p E ] and τ := τ E ∩ M [p],
and thus τ is an O E -lattice in τ E . Note that M i1,j1 E

[p E ]⊗ E Q p is a direct summand of (3.5.48) where Π runs over a subset of automorphic representations in (3.5.48). The same argument as in the paragraph above (4.5.7) of [START_REF] Herzig | On mod p local-global compatibility for GL 3 in the ordinary case[END_REF] using Cebotarev density shows us that the local component Π v of each Π occurring in this direct summand does not depend on Π.

By the definition of τ , we obtain an injection

τ ⊗ O E F → (M [p]) ⊗ O E F = M F [m r ] ( 3
JH soc G(Fp) (τ ⊗ O E F) ⊆ {F (µ ), F (µ ,i1,j1 )}.
Hence by Corollary 3.5.41 we know that

S i1,j1 • S i1,j1 1 (τ ⊗ O E F) U (Fp),µ i 1 ,j 1 1 = 0. (3.5.51)
On the other hand, we have the following equality by Proposition 3.5.25

S i1,j1, • S i1,j1, 1 • (Ξ n ) j1-i1-1 v i1,j1 = κ i1,j1 i0-1 k=j0+1 ψ i1+1+k (p) p (i 0 +j 0 )(i 0 -j 0 -1) 2 S i1,j1 • S i1,j1 1 v i1,j1 .
(3.5.52) By taking mod p reduction of (3.5.52) we deduce from (3.5.49) that

S i1,j1, • S i1,j1, • (Ξ n ) j1-i1-1 v i1,j1 = ε i1,j1 P i1,j1 (b n-1 , • • • , b 0 ) • FL i0,j0 n (r| G Fw ) • S i1,j1 • S i1,j1 1 v i1,j1 .
This equation together with (3.5.51) finishes the proof.

Corollary 3.5.53. Keep the notation of Theorem 3.5.44 and assume that each assumption in Theorem 3.5.44 holds for all (i 0 , j 0 ) such that 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1. Assume further that M i1,j1 is free over T i1,j1 for all pair (i 1 , j 1 ) (c.f. Remark 3.5.47).

Then the structure of S(U v , V )[m r ] as a admissible smooth F-representation of G(Q p ) determines ρ 0 up to isomorphism.

Proof. We follow the notation in Section 3.4 of [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF]. As ρ 0 is ordinary, we can view it as a morphism

ρ 0 : G Qp → B(F) ⊆ G(F)
where B (resp. G) is the dual group of B (resp. G). The local class field theory gives us a bijection between smooth characters of Q × p and the smooth characters of the Weil group of Q p in characteristic zero. This bijection restricts to a bijection between smooth characters of Q × p and smooth characters of Gal(Q p /Q p ) both with values in O × E . Taking mod p reduction and then taking products we reach a bijection between smooth F-characters of T (Q p ) and Hom Gal(Q p /Q p ), T (F) . We can therefore define χ ρ 0 as the character of T (Q p ) corresponding to the composition

χ ρ 0 : Gal(Q p /Q p ) → B(F) T (F).
In [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF], a closed subgroup C ρ 0 ⊆ B (at the beginning of section 3.2) and a subset W ρ 0 ((2) before Lemma 2.3.6) of W is defined.

As we are assuming that ρ 0 is maximally non-split, we observe that C ρ 0 = B and W ρ 0 = {1} in our case. Therefore by the definition of Π ord (ρ 0 ) in [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF] before Definition 3.4.3, we know that it is indecomposable with socle Ind

G(Qp) B -(Qp) χ ρ 0 • (ω -1 • θ)
where θ ∈ X(T ) is a twist character defined after Conjecture 3.1.2 in [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF] which can be chosen to be η in our notation. Then as a Corollary of Theorem 4.4.7 in [START_REF] Breuil | Ordinary representations of G(Q p ) and fundamental algebraic representations[END_REF], we deduce that S(U v , V )[m r ] determines χ ρ 0 and hence χ ρ 0 . Now, we know that ρ 0 is determined by the Fontaine-Laffaille parameters {FL i0,j0 n (ρ 0 ) ∈ P 1 (F) | 0 ≤ i 0 < i 0 + 1 < j 0 ≤ n -1} and χ ρ 0 , up to isomorphism. Our conclusion thus follows from Theorem 3.5.44 and Remark 3.5.47.

Chapter 4

Dilogarithm and higher L -invariants for GL 3 (Q p )

Introduction

Let p be a prime number and F an imaginary quadratic extension of Q such that p splits in F . We fix a unitary algebraic group G over Q which becomes GL n over F and such that G(R) is compact and G is split at all places of F above p. Then to each finite extension E of Q p and to each prime-to-p level U p in G(A ∞,p Q ), one can associate the Banach space of p-adic automorphic forms S(U p , E). One can also associate with U p a set of finite places D(U p ) of Q and a Hecke algebra T(U p ) which is the polynomial algebra freely generated by Hecke operators at places of F lying above D(U p ). In particular, the commutative algebra T(U p ) acts on S(U p , E) and commutes with the action of

G(Q p ) ∼ = GL n (Q p ) coming from translations on G(A ∞ Q ). If ρ : Gal(F /F ) → GL n (E)
is a continuous irreducible representation, one considers the associated Hecke isotypic subspace S(U p , E)[m ρ ], which is a continuous admissible representation of G(Q p ) ∼ = GL n (Q p ) over E, or its locally Q p -analytic vectors S(U p , E)[m ρ ] an , which is an admissible locally Q p -analytic representation of GL n (Q p ). We fix w p a place of F above p and it is widely wished that S(U p , E)[m ρ ] (and its subspace S(U p , E)[m ρ ] an as well) determines and depends only on ρ p := ρ| Gal(Fw p /Fw p ) . The case n = 2 is well-known essentially due to various results in [START_REF] Colmez | Représentations de GL 2 (Q p ) et ( , Γ)-modules[END_REF], [Eme]. The case n ≥ 3 is much more difficult and only some partial results are known. We are particularly interested in the case when the subspace of locally algebraic vectors S(U p , E)[m ρ ] alg S(U p , E)[m ρ ] is non-zero, which implies that ρ p is potentially semi-stable. Certain cases when n = 3 and ρ p is semi-stable and non-crystalline have been studied in [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF]. We are going to continue their work and obtain some interesting relation between results in [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF], [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] and previous results in [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] which involve the p-adic dilogarithm function.

We use the notation λ ∈ X(T ) + for a weight λ = (λ 1 , λ 2 , λ 3 ) (of the diagonal split torus T of GL 3 ) which is dominant with respect to the upper-triangular Borel subgroup B and hence satisfies λ 1 ≥ λ 2 ≥ λ 3 . Given two locally analytic representations V, W of GL 3 (Q p ), we use the shorten notation V W (resp. the shorten notation V W ) for a locally analytic representation determined by a non-zero (resp. possibly zero) element in Ext 1 GL3(Qp) (W, V ).

Theorem 4.1.1. [Proposition 4.6.8, Proposition 4.6.29] For each choice of λ ∈ X(T ) + and L 1 , L 2 , L 3 ∈ E, there exists a locally analytic representation Σ min (λ, L 1 , L 2 , L 3 ) of GL 3 (Q p ) of the form:

St an 3 (λ) v an P1 (λ) C s1,s1 L(λ) ⊗ E v ∞ P2 v an P2 (λ) C s2,s2 L(λ) ⊗ E v ∞ P1 L(λ) L(λ) (4.1.2)
where St an 3 (λ), v an P1 (λ), v an P2 (λ), L(λ) and C * w ,w for w, w ∈ {s 1 , s 2 , s 1 s 2 , s 2 s 1 } and * ∈ {∅, 1, 2} are various explicit locally analytic representations defined in Section 4.2.3. Moreover, different choices of L 1 , L 2 , L 3 ∈ E give non-isomorphic representations.

We will see in Lemma 4.6.47 and (4.6.55) that Σ min (λ, L 1 , L 2 , L 3 ) is the minimal locally analytic representation that involves p-adic dilogarithm, hence explains the notation 'min'. We also construct a locally analytic representation Σ min,+ (λ, L 1 , L 2 , L 3 ) of the form St an 3 (λ)

v an P1 (λ) C s1,s1 L(λ) ⊗ E v ∞ P2 v an P2 (λ) C s2,s2 L(λ) ⊗ E v ∞ P1 L(λ) L(λ) C 1 s2s1,s2s1 C 1 s1s2,s1s2 C 2 s1,s1s2 C 2 s2,s2s1
which contains and is uniquely determined by Σ min (λ, L 1 , L 2 , L 3 ). (iii) ρ p is semi-stable with Hodge-Tate weights {k 1 > k 2 > k 3 } such that N 2 = 0;

(iv) ρ p is non-critical in the sense of Remark 6.1.4 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF];

(v) only one automorphic representation contributes to S(U p , E)[m ρ ] alg .

Then there exists a unique choice of L 1 , L 2 , L 3 ∈ E such that S(U p , E)[m ρ ] an contains (copies of ) the locally analytic representation

Σ min,+ (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det where λ = (λ 1 , λ 2 , λ 3 ) = (k 1 -2, k 2 -1, k 3
) and α ∈ E × is determined by the Weil-Deligne representation WD(ρ p ) associated with ρ p . Moreover, we have

Hom GL3(Qp) Σ min,+ (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] ∼ -→ Hom GL3(Qp) L(λ) ⊗ E St ∞ 3 ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] . (4.1.4)
The assumptions of our Theorem 4.1.3 are the same as that of Theorem 1.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. We do not attempt to obtain any explicit relation between L 1 , L 2 , L 3 ∈ E and ρ p , which is similar in flavor to Theorem 1.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. On the other hand, Theorem 7.52 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] does care about the explicit relation between invariants of the locally analytic representation associated with ρ p , under further technical assumptions such as ρ p is ordinary with consecutive Hodge-Tate weights and has an irreducible mod p reduction but without assuming our condition (v). The improvement of our Theorem 4.1.3 upon Theorem 1.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] will be explained in Section 4.1.2. One can naturally wish that there is a common refinement or generalization of our Theorem 4.1.3 and Theorem 7.52 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] by removing as many technical assumptions as possible.

Remark 4.1.5. It is actually possible to construct a locally analytic representation Σ max (λ, L 1 , L 2 , L 3 ) of GL 3 (Q p ) containing Σ min,+ (λ, L 1 , L 2 , L 3 ) which is characterized by the fact that it is maximal (for inclusion) among the locally analytic representations V satisfying the following conditions:

(i) soc GL3(Qp) (V ) = V alg = L(λ) ⊗ E St ∞ 3 ;
(ii) each constituent of V is a subquotient of a locally analytic principal series where V alg is the subspace of locally algebraic vectors in V . Moreover, one can use an immediate generalization of the arguments in the proof of Theorem 4.1.3 (and thus of Theorem 1.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]) to show that

Hom GL3(Qp) Σ max (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] ∼ -→ Hom GL3(Qp) L(λ) ⊗ E St ∞ 3 ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] . (4.1.6)
We can also show that

Σ max (λ, L 1 , L 2 , L 3 )/L(λ) ⊗ E St 3 is independent of the choice of L 1 , L 2 , L 3 ∈ E, which is compatible with the fact that Σ min, * (λ, L 1 , L 2 , L 3 )/L(λ) ⊗ E St 3
is independent of the choice of L 1 , L 2 , L 3 ∈ E for each * ∈ {∅, +} as mentioned in Remark 4.6.58. However, the full construction of Σ max (λ, L 1 , L 2 , L 3 ) is lengthy and technical and thus we decided not to put it in the present article.

Derived object and dilogarithm

We consider the bounded derived category

D b Mod D(GL3(Qp),E)
associated with the abelian category Mod D(GL3(Qp),E) of abstract modules over the algebra

D(GL 3 (Q p ), E) of locally Q p -analytic distributions on GL 3 (Q p ). An object Σ(λ, L ) ∈ D b Mod D(GL3(Qp),E)
(one should not confuse this notation Σ(λ, L ) borrowed directly from [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] with our notation Σ + (λ, L ) before Lemma 4.6.18) has been constructed in [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and plays a key role in Theorem 1.2 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. An interesting feature of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] is the appearance of the p-adic dilogarithm function in the construction of Σ(λ, L ) in Definition 5.19 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. Roughly, the object Σ(λ, L ) was constructed from the choice of an element in Ext 2 GL3(Qp),λ L(λ), Σ(λ, L 1 , L 2 ) together with general formal arguments in triangulated categories (c.f. Proposition 3.2 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]). In particular, Σ(λ, L ) fits into the following distinguished triangle:

F λ -→ Σ(λ, L ) -→ Σ(λ, L , L ) [-1] +1 --→
as illustrated in (5.99) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. However, it was not clear in [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] whether there is an explicit complex [C • ] of locally analytic representations of GL 3 (Q p ) such that the object

D ∈ D b Mod D(GL3(Qp),E) associated with C -• satisfies D ∼ = Σ(λ, L ) ∈ D b Mod D(GL3(Qp),E) .
Although our notation are slightly different from [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] in the sense that the notation Σ(λ, L , L ) (resp. the notation F λ ) is replaced with Σ(λ, L 1 , L 2 ) (resp. with L(λ)), we show that Theorem 4.1.7. [Proposition 4.6.36, (4.2.28) and Lemma 4.2.37] The complex

L(λ) ⊗ E v ∞ P3-i L(λ) -→ Σ ,+ i (λ, L 1 , L 2 , L 3 ) (4.1.8)
maps to the object Σ(λ, L ) in the derived category where

L(λ) ⊗ E v ∞ P3-i L(λ) is the unique non- split extension of L(λ) by L(λ) ⊗ E v ∞ P3-i thanks to Proposition 4.4.1, Σ ,+ i (λ, L 1 , L 2 , L 3 ) is the locally analytic subrepresentation of Σ min (λ, L 1 , L 2 , L 3 ) of the form St an 3 (λ) v an Pi (λ) C si,si L(λ) ⊗ E v ∞ P3-i v an P3-i (λ) C s3-i,s3-i L(λ)
and the invariants L 1 , L 2 , L 3 ∈ E are determined by the formula

L 1 = -L , L 2 = -L , L 3 = γ(L - 1 2 L L )
with the constant γ ∈ E × defined in Lemma 4.2.34.

Remark 4.1.9. Strictly speaking, the complex (4.1.8) realizes an object in D b Mod D(GL3(Qp),E) characterized by an element in

Ext 2 GL3(Qp),λ L(λ), Σ ,+ (λ, L 1 , L 2 )
due to formal arguments from Proposition 3.2 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. However, we can prove that there is a canonical isomorphism

Ext 2 GL3(Qp),λ L(λ), Σ(λ, L 1 , L 2 ) ∼ -→ Ext 2 GL3(Qp),λ L(λ), Σ ,+ (λ, L 1 , L 2 )
and hence we can equally say that (4.1.8) realizes Σ(λ, L ) for a suitable normalization of notation as Σ(λ, L ) has been constructed by choosing a non-zero element in Ext 2 GL3(Qp),λ L(λ), Σ(λ, L , L ) via Proposition 3.2 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. Note that we have

Σ(λ, L , L ) ∼ = Σ(λ, L 1 , L 2 )
by (4.2.27).

Higher

L -invariants for GL 3 (Q p )
It follows from (4.6.55) and (4.6.57) that Σ min,+ (λ, L 1 , L 2 , L 3 ) can be described more precisely by the following picture:

L(λ) ⊗ E St ∞ 3 C 2 s1,1 C 1 s2s1,1 C 2 s2s1,1 L(λ) ⊗ E v ∞ P1 C 1 s2,1 C s1,s1 L(λ) ⊗ E v ∞ P2 C 2 s2,1 C 1 s1s2,1 C 2 s1s2,1 L(λ) ⊗ E v ∞ P2 C 1 s1,1 C s2,s2 L(λ) ⊗ E v ∞ P1 L(λ) 1 L(λ) 2 C s1s2s1,1 C 1 s2s1,s2s1 C 1 s1s2,s1s2 C 2 s1,s1s2 C 2 s2,s2s1
and therefore contains a unique subrepresentation of the form

L(λ) ⊗ E St ∞ 3 C 2 s1,1 C 1 s2s1,1 L(λ) ⊗ E v ∞ P1 C s1,s1 L(λ) ⊗ E v ∞ P2 C 2 s2,1 C 1 s1s2,1 L(λ) ⊗ E v ∞ P2 C s2,s2 L(λ) ⊗ E v ∞ P1 C 1 s2s1,s2s1 C 1 s1s2,s1s2 C 2 s1,s1s2 C 2 s2,s2s1
which is denoted by

L(λ) ⊗ E St ∞ 3 Π 1 (k, D) Π 2 (k, D) (4.1.10) in Theorem 1.1 of [Bre17]. It follows from Theorem 1.2 of [Bre17] that dim E Ext 1 GL3(Qp),λ Π i (k, D), L(λ) ⊗ E St ∞ 3 = 3
for i = 1, 2, and therefore a locally analytic representation of the form (4.1.10) depends on four invariants. On the other hand, by a computation of extensions of rank one (ϕ, Γ)-modules we know that ρ p depends on three invariants. As a result, Theorem 1.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] predicts that not all representations of the form (4.1.10) can be embedded into S(U p , E) an [m ρ ] for a certain pair of U p and ρ p . This is actually the case as we show that Theorem 4.1.11. [Corollary 4.7.17] If a locally analytic representation Π of the form (4.1.10) can be embedded into S(U p , E) an [m ρ ] for a certain pair of U p and ρ p , then it can be embedded into

Σ min,+ (λ, L 1 , L 2 , L 3 )
for a unique choice of L 1 , L 2 , L 3 ∈ E determined by Π.

Sketch of content

Section 4.2 recalls various well-known facts around locally analytic representations and our notation for a family of specific irreducible subquotients of locally analytic principal series to be used in the rest of the article. We emphasize that our definition of various Ext-groups follows [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] closely and the only difference is that we use the dual notation compared to that of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. We also recall the p-adic dilogarithm function from Section 5.3 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] which is part of the main motivation of this article to relate [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] with [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF]. Section 4.3 proves a crucial fact (Proposition 4.3.14) on the non-existence of locally analytic representations of GL 2 (Q p ) of a certain specific form using arguments involving infinitesimal characters of locally analytic representations. We learn such arguments essentially from Y. Ding.

Section 4.4 is a collection of various computational results necessary for the applications in Section 4.6. These computations essentially make use of the formula in Section 5.2 and 5.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF].

Section 4.5 serves as the preparation of Section 4.6 for the construction of Σ min (λ, L 1 , L 2 , L 3 ). It makes full use of the computational results from Section 4.4 to compute the dimension of various more complicated Ext-groups to be crucially used in various important long exact sequences in Section 4.6(c.f. Lemma 4.6.1 and Proposition 4.6.8).

Section 4.6 finishes the construction of Σ min (λ, L 1 , L 2 , L 3 ) as well as Σ min,+ (λ, L 1 , L 2 , L 3 ). Moreover, the construction of Σ min (λ, L 1 , L 2 , L 3 ) leads naturally to the construction of an explicit complex as in Theorem 4.1.7 that realizes the derived object Σ(λ, L ) constructed in [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF].

Section 4.7 finishes the proof of Theorem 4.7.5 by directly mimicking arguments from the proof of Theorem 6.2.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. In particular, we give a purely representation theoretic criterion for a representation of the form (4.1.10) to embed into completed cohomology as mentioned in Theorem 4.1.11.

Acknowledgement

The author expresses his gratefulness to Christophe Breuil for introducing the problem of relating [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] with [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] and especially for his interest on the role played by the p-adic dilogarithm function. The author also benefited a lot from countless discussions with Y. Ding especially for Section 4.3 of this article. Finally, the author thanks B. Schraen for his beautiful thesis which improved the author's understanding on the subject.

Preliminary

Locally analytic representations

In this section, we recall the definition of some well-known objects in the theory of locally analytic representations of p-adic reductive groups.

We fix a locally Q p -analytic group H and denote the algebra of locally Q p -analytic distribution with coefficient E on H by D(H, E), which is defined as the strong dual of the locally convex Evector space C an (H, E) consisting of locally Q p -analytic functions on H. We use the notation Rep H,E generated by the irreducible objects inside the image of F H P when P varies over all possible parabolic subgroups of H. Here we say a full subcategory is generated by a family of objects if it is the minimal full subcategory that contains these objects and is stable under extensions and taking subquotients. In particular, all objects in Rep OS H,E have finite length.

Formal properties

In this section, we recall and prove some general formal properties of locally analytic representations of p-adic reductive groups. We fix a split p-adic reductive group H and a parabolic subgroup P of H. We use the notation N for the unipotent radical of P and fix a Levi subgroup L of P . Proof. This follows directly from our definition of Ext k and H k in Section 4.2.1 for k ≥ 0, the original dual version in (44) and (45) of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF].

We fix a Borel subgroup B ⊆ H together with its opposite Borel subgroup B. We fix an irreducible object M ∈ O b alg . We choose a parabolic subgroup P ⊆ H such that P is maximal among all the parabolic subgroups Q ⊆ H such that M ∈ O q alg where q is the Lie algebra of the opposite parabolic subgroup Q associated with Q. We fix a smooth irreducible representation π ∞ of L and a smooth character δ of H. Then we know that [START_REF] Orlik | On Jordan-Hölder series of some locally analytic principal series representations[END_REF]

F H P (M, π ∞ ) ⊗ E δ ∼ = F H P (M, π ∞ ⊗ E δ| L ) (4.2.3)
for each irreducible object

F H P (M, π ∞ ) ∈ Rep OS H,E .
Proof. The functor -⊗ E δ is clearly an equivalence of category from Rep la H,E to itself with quasi-inverse given by -⊗ E δ -1 .

It is sufficient to prove the formula (4.2.3) to finish the proof. First of all, we notice by formal reason (equivalence of category) that F H P (M, π ∞ ) ⊗ E δ is an irreducible object in Rep la H,E since F H P (M, π ∞ ) is. We use the notation n (resp. h) for the Lie algebra associated with the unipotent radical N of the opposite parabolic subgroup P of P (resp. for the Lie algebra associated with H). We define M L as the (finite dimensional) algebraic representation of L whose dual is isomorphic to M n as a representation of l and note that we have a surjection

U (h) ⊗ U (p) M n M
where U (h) is the universal enveloping algebra of h. We observe that N acts trivially on δ, and therefore we have

H 0 N, F H P (M, π ∞ ) ⊗ E δ ∼ = H 0 N, F H P (M, π ∞ ) ⊗ E δ| L M L ⊗ E π ∞ ⊗ E δ| L
which induces by Lemma 4.2.1 a non-zero morphism

F H P (M, π ∞ ) ⊗ E δ → Ind H P (M L ⊗ E π ∞ ⊗ E δ| L ) an ∼ = F H P (U (h) ⊗ U (p) M n , π ∞ ⊗ E δ| L ). (4.2.4)
We finish the proof by the fact that F H P (M, π ∞ ) ⊗ E δ is irreducible and that

F H P (M, π ∞ ⊗ E δ| L ) ∼ = soc H F H P (U (h) ⊗ U (p) M n , π ∞ ⊗ E δ| L ) .
due to Corollary 3.3 of [START_REF] Breuil | Vers le socle localement analytique pour GL n I[END_REF].

Some notation

In this section, we are going to recall some standard notation for the p-adic reductive groups GL 2 (Q p ) and GL 3 (Q p ) as well as notation for some locally analytic representations of these groups. We denote the lower-triangular Borel subgroup (resp. the diagonal maximal split torus) of GL 2/Qp by B 2 (resp. by T 2 ) and the unipotent radical of B 2 by N GL2 . We use the notation s for the non-trivial element in the Weyl group of GL 2 . We fix a weight ν ∈ X(T 2 ) of GL 2 of the following form ν = (ν 1 , ν 2 ) ∈ Z 2 which corresponds to an algebraic character of T 2 (Q p ) δ T2,ν := a 0 0 b → a ν1 b ν2 .

We denote the upper-triangular Borel subgroup by B 2 . If ν is dominant with respect to B 2 , namely if ν 1 ≥ ν 2 , we use the notation L GL2 (ν) (resp. L GL2 (-ν)) for the irreducible algebraic representation of GL 2 (Q p ) with highest weight ν (resp. -ν) with respect to the positive roots determined by B 2 (resp. B 2 ). In particular, L GL2 (ν) and L GL2 (-ν) are the dual of each other. We use the shorten notation standard maximal parabolic subgroups of GL 3 with unipotent radical N 1 and N 2 respectively, and the notation P i for the opposite parabolic subgroup of P i for i = 1, 2. We set L i := P i ∩ P i and set s i for the simple reflection in the Weyl group of L i for each i = 1, 2. In particular, the Weyl group W of GL 3 can be lifted to a subgroup of GL 3 with the following elements {1, s 1 , s 2 , s 1 s 2 , s 2 s 1 , s 1 s 2 s 1 }.

I GL2 B2 (χ T2 ) := Ind
The group W acts on X(T ) via the dot action w • λ := w(λ + (2, 1, 0)) -(2, 1, 0).

We will usually use the shorten notation N i (c.f. Section 4.4) for its set of Q p -points N i (Q p ) if it does not cause any ambiguity. We use the notation M (-λ) for the Verma module in O b alg with highest weight -λ (with respect to B) and simple quotient L(-λ) for each λ ∈ X(T ) (not necessarily dominant). Similarly, we use the notation M i (-λ) for the parabolic Verma module in O pi alg with highest weight -λ with respect to B (c.f. Section 9.4 of [START_REF] Humphreys | Representations of Semisimple Lie Algebras in the BGG Category O[END_REF]). We define L i (λ) as the irreducible algebraic representation of L i (Q p ) with a highest weight λ dominant with respect to B ∩ L i . For example, if λ ∈ X(T ) + , then we know that λ, s i • λ and s i s 3-i • λ are dominant with respect to B ∩ L 3-i for i = 1, 2. We use the following notation for various parabolic inductions

I GL3 B (χ) := Ind
B(Qp) χ ∞ ∞ ⊗ E L(λ), i GL3 Pi (π i ) := Ind GL3(Qp) Pi(Qp) π ∞ i ∞ ⊗ E L(λ)
for i = 1, 2 if χ = δ T,λ ⊗ E χ ∞ and π i = L i (λ) ⊗ E π ∞ i are locally algebraic where χ ∞ (resp. π ∞ i ) is a smooth representation of T (Q p ) (resp. of L i (Q p )). We will also use similar notation for parabolic induction to Levi subgroups such as I Li B∩Li and i Li B∩Li for i = 1, 2. Then we define the locally analytic (generalized) Steinberg representation as well as the smooth (generalized) Steinberg representation for GL 3 (Q p ) by where 1 3 (resp. 1 Li ) is the trivial representation of GL 3 (Q p ) (resp. of L i (Q p ) for each i = 1, 2). We define the following irreducible smooth representations of L 1 (Q p ):

π ∞ 1,1 := St ∞ 2 ⊗ E 1 π ∞ 1,2 := i GL2 B2 1 ⊗ E | • | -1 ⊗ E | • | π ∞ 1,3 := St ∞ 2 ⊗ E (| • | -1 • det 2 ) ⊗ E | • | 2
and the following smooth representations of L 2 (Q p ):

π ∞ 2,1 := 1 ⊗ E St ∞ 2 π ∞ 2,2 := | • | -1 ⊗ E i GL2 B2 (| • | ⊗ E 1) π ∞ 2,3 := | • | -2 ⊗ E (St ∞ 2 ⊗ E (| • | • det 2 ))
Consequently, we can define the following locally analytic representations for i = 1, 2:

C 1 si,1 := F GL3 P3-i L(-s i • λ), 1 L3-i C 2 si,1 := F GL3 P3-i L(-s i • λ), π ∞ i,1 C 1 sis3-i,1 := F GL3 P3-i L(-s i s 3-i • λ), 1 L3-i C 2 sis3-i,1 := F GL3 P3-i L(-s i s 3-i • λ), π ∞ i,1 C si,si := F GL3 P3-i L(-s i • λ), π ∞ i,2 C sis3-i,si := F GL3 P3-i L(-s i s 3-i • λ), π ∞ i,2 C 1 si,sis3-i := F GL3 P3-i L(-s i • λ), d ∞ P3-i C 2 s1,1 := F GL3 P3-i L(-s i • λ), π ∞ i,3 C 1 sis3-i,sis3-i := F GL3 P3-i L(-s i s 3-i • λ), d ∞ P3-i C 2 sis3-i,1 := F GL3 P3-i L(-s i s 3-i • λ), π ∞ i,3
(4.2.9) where

d ∞ P1 := | • | -1 • det 2 ⊗ E | • | 2 and d ∞ P2 := | • | -2 ⊗ E | • | • det 2 .
We also define C s1s2s1,w := F GL3 B (L(-s 1 s 2 s 1 • λ), χ ∞ w ) (4.2.10) for each w ∈ W where

χ ∞ 1 := 1 T χ ∞ s1 := | • | -1 ⊗ E | • | ⊗ E 1 χ ∞ s2 := 1 ⊗ E | • | -1 ⊗ E | • | χ ∞ s1s2 := | • | -2 ⊗ E | • | ⊗ E | • | χ ∞ s2s1 := | • | -1 ⊗ E | • | -1 ⊗ E | • | 2 χ ∞ s1s2s1 := | • | -2 ⊗ E 1 ⊗ E | • | 2
As one can write out O b alg explicitly for each parabolic subgroup P ⊆ GL 3 , we notice that the representations considered in (4.2.9) and (4.2.10) are all irreducible objects inside Rep OS GL3(Qp),E according to the main theorem of [START_REF] Orlik | On Jordan-Hölder series of some locally analytic principal series representations[END_REF]. We use the notation Ω for the set whose elements are listed as the following: on the open subgroup 1 + pZ p of Z × p and then extended to Q × p by the condition log 0 (p) = log 0 (ζ) = 0 for each root of unity ζ. We also recall the p-adic valuation function val p : Q × p → Z satisfying | • | = p -valp(•) (and in particular val p (p) = 1). We notice that {log 0 , val p } forms a basis of the two dimensional E-vector space Hom cont Q × p , E .

L(λ) L(λ) ⊗ E v ∞ P1 L(λ) ⊗ E v ∞ P2 L(λ) ⊗ E St ∞ 3 C 1 s1,1 C 2 s1,1 C 1 s2,1 C 2 s2,1 C 1 s1s2,1 C 2 s1s2,1 C 1 s2s1,1 C 2 s2s1,1 C 1 s1,s1s2 C 2 s1,s1s2 C 1 s2,s2s1 C 2 s2,s2s1 C 1 s1s2,s1s2 C 2 s1s2,s1s2
We define log L := log 0 -L val p for each L ∈ E and consider the following two dimensional locally analytic representation of

Q × p V L : Q × p → B 2 (E), a → 1 log L (a) 0 1
and therefore soc

Q × p (V L ) = cosoc Q × p (V L ) = 1 (4.2.19)
where 1 is the notation for the trivial character of Q × p . We notice that Ext 1 As a result, we can consider the parabolic induction

Q × p (1, 1) ∼ = Hom cont Q × p , E ,
I GL2 B2 (V L ⊗ E δ T2,ν )
which naturally fits into an exact sequence

I GL2 B2 (δ T2,ν ) → I GL2 B2 (V L ⊗ E δ T2,ν ) I GL2 B2 (δ T2,ν ). ( 4

.2.21)

Then we define Σ GL2 (ν, L ) as the subrepresentation of I GL2 B2 (V L ⊗ E δ T2,ν ) /L GL2 (ν) with cosocle L GL2 (ν). It follows from (the proof of) Theorem 3.14 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] where Σ(λ, L , L ) is the locally analytic representation defined in Definition 5.12 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] using the element (c 2,log + L c 2,val , c 1,log + L c 1,val ) in Ext 1 GL3(Qp),λ v an P1 (λ) ⊕ v an P2 (λ), St an 3 (λ) . Remark 4.2.29. The appearance of a sign in (4.2.28) (which is an issue of normalization) is essentially due to Remark 3.1 of [START_REF] Ding | Simple L-invariants for GL n[END_REF], which implies that our invariants L 1 and L 2 can be identified with Fontaine-Mazur L -invariants of the corresponding Galois representation via local-global compatibility.

We have a canonical morphism by (5.26) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] κ : Ext 2 T (Qp),0 (1 T , 1 T ) → Ext 2 GL3(Qp),λ L(λ), St an 3 (λ) . as in (5.36) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] which is also a locally analytic function over Q p \ {0, 1} and is independent of the choice of L ∈ E. Note by our definition that

D L -D 0 = L 2 d.
It follows from Theorem 7.2 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] that {D 0 , d} can be identified with a basis of Ext for some µ = λ H , which contradicts the fact (4.3.5). Hence we conclude that j 1 (z) -j 2 • θ(z) ∈ U (h) • n H and in particular j 1 (z) = j 2 • θ(z)

on V N H •
for each z ∈ Z(U (h)). Hence we deduce that U (t h ) W H acts on V N H • via a character. We note by the definition of J B H (c.f. [START_REF] Emerton | Jacquet modules of locally analytic representations of p-adic reductive groups I: constructions and first properties[END_REF]) that we have a T + H -equivariant embedding

J B H (V ) → V N H • (4.3.8)
where T + H is a certain submonoid of T H containing an open compact subgroup. As a result, (4.3.8) is also U (t h )-equivariant and thus U (t h ) W H acts on J B H (V ) via a character which finishes the proof.

We set H = GL 2 (Q p ), B H = B 2 and B H = B 2 in the rest of this section. The idea of the following lemma which is closely related to Lemma 3.20 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF], owes very much to Y.Ding.

Lemma 4.3.9. A locally analytic representation of either the form

L GL2 (ν) ⊗ E St ∞ 2 I(s • ν) L GL2 (ν) L GL2 (ν) ⊗ E St ∞ 2 (4.3.10)
or the form

L GL2 (ν) I(s • ν) L GL2 (ν) ⊗ E St ∞ 2 L GL2 (ν) (4.3.11)
does not have an infinitesimal character.

Proposition 4.3.14. We have

Ext 1 GL2(Qp) L GL2 (ν) ⊗ E St ∞ 2 L GL2 (ν) , Σ + 2 (ν, L ) = 0.
Proof. Assume on the contrary that V is a representation given by a certain non-zero element inside Ext 1 GL2(Qp)

L GL2 (ν) ⊗ E St ∞ 2 L GL2 (ν) , Σ + 2 (ν, L ) .
We deduce that V has both a central character and an infinitesimal character from Lemma 4.3.3 and the fact Hom GL2(Qp) L GL2 (ν)

⊗ E St ∞ 2 L GL2 (ν) , Σ + 2 (ν, L ) = 0.
Note that we have

Ext 1 GL2(Qp) (L GL2 (ν) ⊗ E St ∞ 2 , I(s • ν)) = Ext 1 GL2(Qp) (L GL2 (ν), I(s • ν)) = 0, dim E Ext 1 GL2(Qp) L GL2 (ν), L GL2 (ν) ⊗ E St ∞ 2 = 1
and dim E Ext 1 GL2(Qp) L GL2 (ν), I(s • ν) = 1 by a combination of Lemma 3.13 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] with Lemma 4.2.1, and thus V has a subrepresentation of one of the three following forms

(i) L GL2 (ν) ⊗ E St ∞ 2 L GL2 (ν) ⊗ E St ∞ 2 ; (ii) L GL2 (ν) ⊗ E St ∞ 2 I(s • ν) L GL2 (ν) L GL2 (ν) ⊗ E St ∞ 2 ; (iii) L GL2 (ν) ⊗ E St ∞ 2 I(s • ν) L GL2 (ν) I(s • ν) L GL2 (ν) ⊗ E St ∞ 2 L GL2 (ν) .
In the first case, we know from Proposition 4.7 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and the main result of [START_REF] Orlik | On extensions of generalized Steinberg representations[END_REF] that

Ext 1 GL2(Qp),ν L GL2 (ν) ⊗ E St ∞ 2 , L GL2 (ν) ⊗ E St ∞ 2 = 0
and therefore this case is impossible due to the existence of central character for V (and hence for its subrepresentations). In the second case, we deduce from Lemma 4.3.9 a contradiction as V has an infinitesimal character. In the third case, we thus know that V has a quotient representation of the form L GL2 (ν)

I(s • ν) L GL2 (ν) ⊗ E St ∞ 2 L GL2 (ν)
which can not have an infinitesimal character due to Lemma 4.3.9, a contradiction again. Hence we finish the proof.

L GL2 (ν) ⊗ E St ∞ 2 L GL2 (ν) , I(s • ν) L GL2 (ν) ⊗ E St ∞ 2 I(s • ν) = 0.
or of the form

C si,si C 1 s3-isi,s3-isi L(λ) ⊗ E v ∞ P3-i
C 2 si,sis3-i .

Proof. We only prove the first statement as the second is similar. It follows from Proposition 4.4.2 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] that there exists a unique representation of the form

C 2 si,1 C 1 s3-isi,1 L(λ) ⊗ E v ∞ Pi C si,si
but it is not proven there whether its quotient

C 1 s3-isi
L(λ) ⊗ E v ∞ Pi C si,si
which contradicts the first vanishing in Lemma 4.4.31, and thus we finish the proof. Proposition 4.6.8. We have dim E Ext 1 GL3(Qp),λ W 0 , Σ + (λ, L 1 , L 2 ) = 3.

Proof. The short exact sequence

L(λ) ⊗ E v ∞ P2 ⊕ v ∞ P1 → W 0 L(λ)
induces a long exact sequence Ext 1 GL3(Qp),λ L(λ),

Σ + (λ, L 1 , L 2 ) → Ext 1 GL3(Qp),λ W 0 , Σ + (λ, L 1 , L 2 ) → Ext 1 GL3(Qp),λ L(λ) ⊗ E v ∞ P2 ⊕ v ∞ P1 , Σ + (λ, L 1 , L 2 ) → Ext 2 GL3(Qp)
,λ L(λ), Σ + (λ, L 1 , L 2 ) (4.6.9)

and thus we have

dim E Ext 1 GL3(Qp),λ (W 0 , Σ + (λ, L 1 , L 2 )) ≥ dim E Ext 1 GL3(Qp),λ (L(λ), Σ + (λ, L 1 , L 2 ))+dim E Ext 1 GL3(Qp),λ (L(λ)⊗ E v ∞ P2 ⊕ v ∞ P1 , Σ + (λ, L 1 , L 2 )) -dim E Ext 2
GL3(Qp),λ (L(λ), Σ + (λ, L 1 , L 2 )) = 1 + 4 -2 = 3 (4.6.10) due to Lemma 4.5.7 and Lemma 4.5.13, which finishes the proof by combining with Lemma 4.6.1.

We define Σ (λ, L 1 , L 2 ) as the unique non-split extension of L(λ) by Σ(λ, L 1 , L 2 ) (c.f. Lemma 4.2.35) and then set Σ ,+ (λ, L 1 , L 2 ) to be the amalgamate sum of Σ (λ, L 1 , L 2 ) and Σ + (λ, L 1 , L 2 ) over Σ(λ, L 1 , L 2 ). Hence Σ (λ, L 1 , L 2 ) has the form generates a line in Ext 2 GL3(Qp),λ L(λ), Σ + (λ, L 1 , L 2 ) for each L ∈ E. We define Σ + i (λ, L 1 , L 2 , L i ) as the representation represent by the preimage of

ι 1 (D 0 ) + L i κ(b 1,valp ∧ b 2,valp ) in Ext 1 GL3(Qp),λ L(λ) ⊗ E v ∞ Pi , Σ + (λ, L 1 , L 2 )
via (4.6.16) for i = 1, 2. Then we define Σ + (λ, L ) as the amalgamate sum of Σ + 1 (λ, L 1 , L 2 , L 1 ) and Σ + 2 (λ, L 1 , L 2 , L 2 ) over Σ + (λ, L 1 , L 2 ), and therefore Σ + (λ, L ) has the form St an 3 (λ)

v an P1 (λ) v an P2 (λ) C s1,s1 C s2,s2 L(λ) ⊗ E v ∞ P2 L(λ) ⊗ E v ∞ P1 .
We define Σ ,+ (λ, L ) as the amalgamate sum of Σ + (λ, L ) and Σ (λ, L 1 , L 2 ) over Σ(λ, L 1 , L 2 ), and thus Σ ,+ (λ, L ) has the form St an 3 (λ)

v an P1 (λ) v an P2 (λ) C s1,s1 C s2,s2 L(λ) L(λ) ⊗ E v ∞ P2 L(λ) ⊗ E v ∞ P1 .
We also need the quotients Σ +, (λ, L ) := Σ + (λ, L )/L(λ) ⊗ E St ∞ 3 , Σ ,+, (λ, L ) := Σ ,+ (λ, L )/L(λ) ⊗ E St ∞ 3 .

Lemma 4.6.18. We have the inequality dim E Ext 1 GL3(Qp),λ L(λ), Σ ,+, (λ, L ) ≤ 1.

Proof. The short exact sequence Σ ,+, (λ, L 1 , L 2 ) → Σ ,+, (λ, L ) is a distinguished triangle. On the other hand, it is easy to see that D i (λ, L 1 , L 2 , L 3 ) fits into the distinguished triangle

L(λ) ⊗ E v ∞ P2 ⊕ v ∞
W 3-i -→ Σ ,+ i (λ, L 1 , L 2 , L 3 ) -→ D i (λ, L 1 , L 2 , L 3 ) +1 --→
and thus we conclude that

D i (λ, L 1 , L 2 , L 3 ) ∼ = D(λ, L 1 , L 2 , L 3 ) ∈ D b Mod D(GL3(Qp),E)
by the uniqueness in Proposition 3.2 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. Hence we finish the proof.

We define Σ min,-(λ, L 1 , L 2 , L 3 ) as the unique subrepresentation of Σ min (λ, L 1 , L 2 , L 3 ) of the form St an 3 (λ)

v an P1 (λ) v an P2 (λ) C s1,s1 C s2,s2 L(λ) ⊗ E v ∞ P2 L(λ) ⊗ E v ∞ P1
We use the notation L(λ) i for copy of L(λ) inside L(λ) ⊕2 corresponding to the one dimensional space M min ∩ M {1,2,i+2} inside M min , and therefore we have a surjection

Σ min (λ, L 1 , L 2 , L 3 ) C 1 s2,1 L(λ) 1 ⊕ C 1 s1,1
L(λ) 2 . (4.6.53)

As a result, the representation Σ min (λ, L 1 , L 2 , L 3 ) has the following form:

St an 3 (λ)

v an P1 (λ) C s1,s1 L(λ) ⊗ E v ∞ P2 v an P2 (λ) C s2,s2 L(λ) ⊗ E v ∞ P1 L(λ) 1 L(λ) 2
.

(4.6.54)

If we clarify the internal structure of St an 3 (λ), v an P1 (λ) and v an P2 (λ) using Lemma 4.2.13, then Σ min (λ, L 1 , L 2 , L 3 ) has the following form:

L(λ) ⊗ E St ∞ 3 C 2 s1,1 C 1 s2s1,1 C 2 s2s1,1 L(λ) ⊗ E v ∞ P1 C 1 s2,1 C s1,s1 L(λ) ⊗ E v ∞ P2 C 2 s2,1 C 1 s1s2,1 C 2 s1s2,1 L(λ) ⊗ E v ∞ P2 C 1 s1,1 C s2,s2 L(λ) ⊗ E v ∞ P1 L(λ) 1 L(λ) 2 C s1s2s1,1
. (4.6.55)

Remark 4.6.56. It is actually possible to show that all the possibly split extensions illustrated in (4.6.55) are non-split. However, the proof is quite technical and not related to the p-adic dilogarithm function, and thus we decided not to include the proof here.

We observe that Σ min (λ, L 1 , L 2 , L 3 ) admits a unique subrepresentation Σ Ext 1 ,-(λ, L 1 , L 2 , L 3 ) of the form

L(λ) ⊗ E St ∞ 3 C 2 s1,1 C 1 s2s1,1 L(λ) ⊗ E v ∞ P1 C s1,s1 L(λ) ⊗ E v ∞ P2 C 2 s2,1 C 1 s1s2,1 L(λ) ⊗ E v ∞ P2 C s2,s2 L(λ) ⊗ E v ∞
which can be uniquely extend to a representation Σ Ext 1 (λ, L 1 , L 2 , L 3 ) of the form:

L(λ) ⊗ E St ∞ 3 C 2 s1,1 C 1 s2s1,1 L(λ) ⊗ E v ∞ P1 C s1,s1 L(λ) ⊗ E v ∞ P2 C 2 s2,1 C 1 s1s2,1 L(λ) ⊗ E v ∞ P2 C s2,s2 L(λ) ⊗ E v ∞ P1 C 1 s2s1,s2s1 C 1 s1s2,s1s2 C 2 s1,s1s2
C 2 s2,s2s1

(4.6.57) according to Section 4.4 and 4.6 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] together with our Lemma 4.4.34. Finally, we define Σ min,+ (λ, L 1 , L 2 , L 3 ) as the amalgamate sum of Σ min (λ, L 1 , L 2 , L 3 ) and Σ Ext 1 (λ, L 1 , L 2 , L 3 ) over Σ Ext 1 ,-(λ, L 1 , L 2 , L 3 ).

Local-global compatibility

We are going to borrow most of the notation and assumptions from Section 6 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. We fix embeddings ι ∞ : Q → C, ι p : Q → Q p , an imaginary quadratic CM extension F of Q and a unitary group G/Q attached to the extension F/Q such that G × Q F ∼ = GL 3 and G(R) is compact. If is a finite place of Q which splits completely in F , we have isomorphisms ι G,w : G(Q ) ∼ -→ G(F w ) ∼ = GL 3 (F w ) for each finite place w of F over . We assume that p splits completely in F , and we fix a finite place w 0 of F dividing p and therefore G(Q p ) ∼ = G(F w0 ) ∼ = GL 3 (Q p ).

We fix an open compact subgroup U p G(A ∞,p Q ) of the form U p = =p U where U is an open compact subgroup of G(Q ). For each finite extension E of Q p inside Q p , we consider the following O E -lattice inside a p-adic Banach space:

S(U p , O E ) := {f : G(Q)\G(A ∞ Q )/U p → O E , f continuous} (4.7.1)
and note that S(U p , E) := S(U p , O E ) ⊗ O E E. The right translation of G(Q p ) on G(Q)\G(A ∞ Q )/U p induces a p-adic continuous action of G(Q p ) on S(U p , O E ) which makes S(U p , E) an admissible Banach representation of G(Q p ) in the sense of [START_REF] Schneider | Banach space representations and Iwasawa theory[END_REF]. We use the notation S(U p , E) alg ⊆ S(U p , E) an following Section 6 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] for the subspaces of locally Q p -algebraic vectors and locally Q p -analytic vectors inside S(U p , E) respectively. Moreover, we have the following decomposition:

S(U p , E) alg ⊗ E Q p ∼ = π (π v0 f ) Up ⊗ Q (π v0 ⊗ Q W p ) (4.7.2)
where the direct sum is over the automorphic representations π of G(A Q ) over C and W p is the Q palgebraic representation of G(Q p ) over Q p associated with the algebraic representation π ∞ of G(R) over C via ι p and ι ∞ . In particular, each distinct π appears with multiplicity one (c.f. the paragraph after (55) of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] for further references). We use the notation D(U p ) for the set of finite places of Q that are different from p, split completely in F and such that U is a maximal open compact subgroup of G(Q ). Then we consider proof of Theorem 4.7.5. We may assume that α = 1 for simplicity of notation thanks to Lemma 4.2.2. According to the Étape 1 and 2 of Section 6.2 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF], we may assume without loss of generality that U p is sufficiently small and it is sufficient to show that there exists a unique choice of L 1 , L 2 , L 3 ∈ E such that Hom GL3(Qp) Σ min,+ (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] = 0.

(4.7.10)

We borrow the notation Π i (k, D) from Theorem 6.2.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. We observe from (4.6.55) that Σ min,+ (λ, L 1 , L 2 , L 3 ) contains a unique subrepresentation Σ Ext 1 (λ, L 1 , L 2 , L 3 ) of the form

L(λ) ⊗ E St ∞ 3 Π 1 (k, D) Π 2 (k, D)
.

(4.7.11) Moreover, Σ min,+ (λ, L 1 , L 2 , L 3 ) is uniquely determined by Σ Ext 1 (λ, L 1 , L 2 , L 3 ) up to isomorphism.

It is known by Étape 3 of Section 6.2 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] that there is at most one choice of L 1 , L 2 , L 3 ∈ E such that Hom GL3(Qp) Σ Ext 1 (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] = 0, and thus there is at most one choice of L 1 , L 2 , L 3 ∈ E such that (4.7.10) holds. As a result, it remains to show the existence of L 1 , L 2 , L 3 ∈ E that satisfies (4.7.10). We notice that Σ min,+ (λ, L 1 , L 2 , L 3 ) admits an increasing filtration Fil • satisfying the following conditions (i) the representations Σ min (λ, L 1 , L 2 , L 3 ) and Σ ,+ (λ, L 1 , L 2 ) (c.f. their definition after Proposition 4.6.8 and Proposition 4.6.29) appear as two consecutive terms of the filtration;

(ii) each graded piece is either locally algebraic or irreducible.

As a result, the only reducible graded pieces of this filtration is the quotient Σ min (λ, L 1 , L 2 , L 3 )/Σ ,+ (λ, L 1 , L 2 ) ∼ = W 0 .

Then we can prove the existence of L 1 , L 2 , L 3 ∈ E satisfying (4. is not locally algebraic, then (4.7.12) is true in this case by part (i) of Proposition 4.7.9. The only locally algebraic graded pieces of the filtration except L(λ)

⊗ E St ∞ 3 are L(λ) ⊗ E v ∞ P1 , L(λ) ⊗ E v ∞
P2 and W 0 . The isomorphism (4.7.12) when the graded piece Gr k equals L(λ) ⊗ E v ∞ P1 or L(λ) ⊗ E v ∞ P2 has been treated in Étape 2 of Section 6.4 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. As a result, it remains to show that Hom GL3(Qp) Σ min (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] ∼ -→ Hom GL3(Qp) Σ ,+ (λ, L 1 , L 2 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] (4.7.13)

  3.5), (3.3.6), and (3.3.7). That is, there exist a framed basis e for M and a framed system of generators f for Fil n-1 M such that Mat e,f (Fil n-1 M), Mat e,f (φ n-1 ), Mat e (N ) are given as in (3.3.5), (3.3.6), and (3.3.7) respectively. Sincek i ≡ k (0) i mod (p -1), we have r i = r (0) i for all i ∈ {0, 1, • • • , n -1}by Lemma 3.3.10), following the notation of Lemma 3.3.10. We start to prove the following claim:

  s ) ≡ 0 modulo (p) for all 0 ≤ t ≤ r (0) p0 -2. Hence, we conclude that x (t) p0,s = 0 for all 0 ≤ t ≤ r (0) p0 -2 as u e N (f s ) ∈ Fil n-1 M p0,0 , which completes the proof.Proposition 3.3.27. Keep the assumptions and notation of Lemma 3.3.21. Assume further that ρ 0 is maximally non-split and satisfies the conditions in (3.3.16).

  3.20) as well as k (0) i . Let M ∈ O E -Mod n-1 dd be a strongly divisible module corresponding to a lattice in a potentially semi-stable representation ρ : G Qp → GL n (E) with Galois type n-1 i=0 ω k (0) i and Hodge-Tate weights {

68CHAPTER 3 .

 3 MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Then every potentially semi-stable lift of ρ 0 with Hodge-Tate weights {-(n -1), -(n -2), • • • , 0} and Galois types n-1 i=0 ω ki

  [x] 1 to denote the only integer satisfying 0 ≤ [x] 1 ≤ p -2 and [x] 1 ≡ x mod (p -1). Given two non-negative integers m and k with m ≥ k, we use the notation c m,k for the binomial number m! (m-k)!k! . We use the notation • for composition of maps and, in particular, composition of elements in the group algebra F p [G(F p )].

78CHAPTER 3 .

 3 MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE

  .4.14) The last computation (3.4.14) follows from the fact that t∈Fp t = 0 if p -1 ; -1 if p -1 | and = 0. (3.4.15) Applying (3.4.14) back to (3.4.13) gives us the result. Lemma 3.4.16. Fix w ∈ W and α 0

  .4.28) for all t ∈ F p . We define Φ α0,+ w := {α ∈ Φ + w | α 0 ≺ α} and Φ α0,- w := Φ + w \ Φ α0,+ w , and we use the notation := ( α ) α∈Φ α 0 ,- w ∈ {0, • • • , p -1} |Φ α 0 ,- w | for a tuple of integers indexed by Φ α0,- w . Given a tuple , we can define Λ( , α 0

  Remark 3.4.38. Theorem 3.4.35 and Corollary 3.4.37 can be generalized to the case when µ * is replaced by any weight lying sufficiently deep in an arbitrary alcove. The crucial points here are the [U (F p ), U (F p )]-invariance of S n (resp. S n ) and that τ (in Corollary 3.4.37) is one of the simplest lattices of π • 0 ⊗ O E E apart from those coming from parabolic inductions from B(F p ).

  (w)+ (w )-(ww ) 2 S ww on ( π • ) I(1) for all w, w ∈ W . Proof. One can quickly reduce the general case to the following two elementary equalities on ( π • ) I(1) : S w • S w = S ww if (ww ) = (w) + (w ) (3.4.52) and S sr • S sr = p for all 1 ≤ r ≤ n -1. (3.4.53)

  For each pair of integers (a, b) with 0 ≤ a, b ≤ p -1, we set J(a, b) := λ∈Fp λ a 1 -λ b . (3.4.55) We also set G(a) := λ∈Fp λ a ξ λ

  .4.57) By Stickelberger's theorem ([Lang] Section 1.2, Theorem 2.1), we know that ord p (G(a)) = 1 -a p -1 and G(a) p-1-a ≡ a! (mod p). (3.4.58) Let r ∈ Z with 1 ≤ r ≤ n -1 and w ∈ W . Given the data µ π = (d 1 , d 2 , • • • , d n ) and tuple k ∈ {0, . . . , p -1} |Φ + w | , we define a tuple k ∈ {0, . . . , p -1} |Φ + w | if (ws r ) < (w); {0, . . . , p -1} |Φ + wsr | if (ws r ) > (w) (3.4.59) 92CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE by

  and D n are polynomials over the entries of A. Given a weight λ ∈ X + (T ), we now introduce an explicit model for the representation H 0 (λ), and then start some explicit calculation. Consider the space of polynomials on G /Fp , which is denoted by O(G). The space O(G) has both a left action and a right action of B induced by right translation and left translation by B on G respectively. The fraction field of O(G) is denoted by M(G).

98CHAPTER 3 .

 3 MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Consider the subspace O(λ) := {f ∈ O(G) | f • b = w 0 λ(b)f ∀b ∈ B},which has a natural left G-action by right translation. As the right action of T on O(G) is semisimple (and normalizes U ), we have a decomposition of algebraic representations of G:

104CHAPTER 3 .

 3 MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE By the Bruhat decomposition in (3.4.93), we have

  4.121) by (3.4.115) for r = n --1, and

  4.122) by (3.4.114) for r = n --1 where i 0 = {n --1, n}. Now assume that (3.4.118) is true for some 1 ≤ ≤ -2. Then by combing (3.4.119), (3.4.120), (3.4.121) and (3.4.122), we have

w0 v 0 where i 1

 01 = {n}. Finally, (3.4.114) for r = n -follows from the equation above together with the definition of Z and the identity (3.4.108). Proof of Proposition 3.4.113. It follows easily from Lemma 3.4.116 and Lemma 3.4.117. Proposition 3.4.124. For each 1 ≤ r ≤ n -2 and 1

∼→

  C for the rest of the paper. As we did for the O Fw -specialization of the dual Weyl modules, we define a finite dimensionalG n (F + ⊗ Q R)-representation σ a ∼ =v|∞ σ a v with C-coefficients. (We refer to [EGH13], Section 7.1.4 for the precise definition of σ a .) 122CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE Lemma 3.5.7 ([EGH13], Lemma 7.1.6). The isomorphism ı : Q

:

  F × → Q × p denote the unique square root of | | 1-n whose composite with ι : Q p ∼ → C takes positive values. Theorem 3.5.8 ([EGH13], Theorem 7.2.1). Let Π be an irreducible

Fw,n j

  by inflation and we define the following Galois type rec(θ) : I Fw → GL n (Q p ) as follows: rec(θ) := r j=1   σ∈Gal(kw,n j /kw)

  Theorem 3.5.12 ([CEGGPS], Theorem 3.7 and [LLL16], Proposition 2.3.4). Let τ : I Qp → GL n (Q p ) be a Galois type. Then there exists a finite dimensional irreducible smooth Q p -representation σ(τ ) of GL n (Z p ) such that if π is any irreducible smooth Q p -representation of GL n (Q p ) then π| GLn(Zp) contains a unique copy of σ(τ ) as a subrepresentation if and only if rec Qp (π)| I Qp ∼ = τ and N = 0 on rec Qp (π).

n

  (c.f. (3.4.63)),κ n (c.f. (3.4.70)), ε * (c.f. (3.4.69)), and P n (c.f. (3.4.68)), whose definitions are completely determined by fixing the data n and (a n-1

  .5.50) as p + E T P = m r . By the assumption (3.5.45) (c.f. Conjecture 3.5.16), we deduce that JH soc G(Fp) (M F [m r ]) ⊆ {F (µ ), F (µ ,i1,j1 )} and therefore by (3.5.50) we have

  Theorem 4.1.3. [Theorem 4.7.5] Assume that p ≥ 5 and n = 3. Assume moreover that (i) ρ is unramified at all finite places of F above D(U p ); (ii) S(U p , E)[m ρ ] alg = 0;

Lemma



  GL2(Qp)B2(Qp) χ T2 an for any locally analytic character χ T2 of T 2 (Q p ) and seti GL2 B2 (χ T2 ) := Ind GL2(Qp) B2(Qp) χ ∞ T2 ∞ ⊗ E L GL2 (ν) if χ T2 = δ T2,ν ⊗ E χ ∞ T2 is locally algebraic where χ ∞ T2 is a smooth character of T 2 (Q p ).Then we define the locally analytic Steinberg representation as well as the smooth Steinberg representation for GL 2 (Q p ) as followsSt an 2 (ν) := I GL2 B2 (δ T2,µ )/L GL2 (ν), St ∞ 2 := i GL2 B2 (1 T2 )/1 2 where 1 2 (resp. 1 T2 ) is the trivial representation of GL 2 (Q p ) (resp. of T 2 (Q p )).We denote the lower-triangular Borel subgroup (resp. the diagonal maximal split torus) of GL 3/Qp by B (resp. by T ) and the unipotent radical of B by N . We fix a weight λ ∈ X(T ) of GL 3 of the following form λ = (λ 1 , λ 2 , λ 3 ) ∈ Z 3 , which corresponds to an algebraic character ofT (Q p ) → a λ1 b λ2 c λ3 .We denote the center of GL 3 by Z and notice that Z(Q p ) ∼ = Q × p . Hence the restriction of δ T,λ to Z(Q p ) gives an algebraic character of Z(Q p ): λ1+λ2+λ3 .We use the shorten notationExt i * ,λ (-, -) := Ext i * ,δ Z,λ (-, -)for * ∈ {T (Q p ), L 1 (Q p ), L 2 (Q p ), GL 3 (Q p )}.In particular, the notation Ext i * ,0 (-, -) means (higher) extensions with the trivial central character. We denote the upper-triangular Borel subgroup of GL 3 by B. If λ is dominant with respect to B, namely if λ 1 ≥ λ 2 ≥ λ 3 , we use the notation L(λ) (resp. L(-λ)) for the irreducible algebraic representation of GL 3 (Q p ) with highest weight λ (resp.-λ) with respect to the positive roots determined by B (resp. B). In particular, L(λ) and L(-λ) are dual of each other. We use the notation P 1

  Pi (π i ) := Ind GL3(Qp) Pi(Qp) π i an if χ is an arbitrary locally analytic character of T (Q p ) and π i is an arbitrary locally analytic representation of L i (Q p ) for each i = 1, 2. Moreover, we use the notation i GL3 B (χ) := Ind GL3(Qp)

  St an 3 (λ) := I GL3 B (δ T,λ )/ I GL3 P1 (L 1 (λ)) + I GL3 P2 (L 2 (λ)) , St ∞ 3 := i GL3 B (1)/ i GL3 P1 (1 L1 ) + i GL3 P2 (1 L2 ) and v an Pi (λ) := I GL3 Pi (L i (λ))/L(λ), v ∞ Pi := i GL3 Pi (1 Li )/1 3

  by a standard fact in (continuous) group cohomology and therefore the set {V L | L ∈ E} exhausts (up to isomorphism) all different two dimensional locally analytic non-smooth E-representations of Q × p satisfying (4.2.19). We observe that V L can be viewed as a representation ofT 2 (Q p ) ∼ = Q × p × Q × pby composing with the mapT 2 (Q p ) → Q × p :

  (4.2.30) Note that we also have Ext 2 T (Qp),0 (1T , 1 T ) ∼ = ∧ 2 Ext 1 T (Qp),0 (1 T , 1 T )by (5.24) of[START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and thus the set{b 1,valp ∧ b 2,valp , b 1,log 0 ∧ b 2,valp , b 1,valp ∧ b 2,log 0 , b 1,log 0 ∧ b 2,log 0 , b 1,valp ∧ b 1,log 0 , b 2,valp ∧ b 2,log 0 } forms a basis of Ext 2 T (Qp),0 (1 T , 1 T ). Itfollows from (5.27) of [Schr11] and (4.2.24) that the set {κ(b 1,valp ∧ b 2,valp ), κ(b 1,log 0 ∧ b 2,valp ), κ(b 1,valp ∧ b 2,log 0 ), κ(b 1,log 0 ∧ b 2,log 0 )} forms a basis of the image of (4.2.30). We recall the p-adic dilogarithm function li 2 : Q p \ {0, 1} → Q p defined by Coleman in [Cole82] and we consider the functionD L (z) := li 2 (z) + 1 2 log L (z)log L (1 -z)as in (5.34) of[START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. We also define d(z) := log L (1 -z)val p (z) -log L (z)val p (1 -z)

2 L

 2 Remark 4.3.15. Note that the argument in Proposition 4.3.14 actually implies thatExt 1 GL2(Qp) L GL2 (ν) ⊗ E St ∞ GL2 (ν) , I(s • ν) L GL2 (ν) I(s • ν) = 0and we can show by the same method that Ext 1 GL2(Qp)

GL3C

  Remark 4.4.36. Our method used in Lemma 4.4.31 and in Lemma 4.4.34 is different from the one due to Y.Ding mentioned in part (ii) of Remark 4.4.3 of[START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. It is not difficult to observe thatdim E Ext 1 -i , C si,si C 1 s3-isi,s3-isi L(λ) ⊗ E v ∞ for i = 1, 2.Similar methods as those used in Proposition 4.4.2 of [Bre17], in Lemma 4.4.31 and in Lemma 4.4.34 also imply the existence of a unique representation of the form s3-isi,s3-i L(λ) ⊗ E St ∞ 3 C 2 si,1 .

  It follows from Lemma 4.2.35, Proposition 4.4.1, (4.5.17) and an easy devissage thatExt 1 GL3(Qp),λ L(λ), Σ (λ, L 1 , L 2 ) = Ext 1 GL3(Qp),λ L(λ), Σ ,+ (λ, L 1 , L 2 ) = 0. (4.6.11) Then we set Σ * , (λ, L 1 , L 2 ) := Σ * (λ, L 1 , L 2 )/L(λ) ⊗ E St ∞ 3for * = {+}, { } and { , +}. It follows from Lemma 4.5.28, (4.5.17) and an easy devissage thatExt 1 GL3(Qp),λ L(λ), Σ , (λ, L 1 , L 2 ) = Ext 1 GL3(Qp),λ L(λ), Σ ,+, (λ, L 1 , L 2 ) = 0.(4.6.12) Lemma 4.6.13. We haveExt 1 GL3(Qp),λ L(λ), Σ (λ, L 1 , L 2 ) = Ext 1 GL3(Qp),λ L(λ), Σ ,+ (λ, L 1 , L 2 ) = 0anddim E Ext 2 GL3(Qp),λ L(λ), Σ (λ, L 1 , L 2 ) = dim E Ext 2 GL3(Qp),λ L(λ), Σ ,+ (λ, L 1 , L 2 ) = 2where Ext 1 GL3(Qp),λ L(λ), L(λ) ⊗ E v ∞ Pi is one dimensional by Proposition 4.4.1. We recall from the proof of Lemma 4.5.13 that there is a canonical isomorphismExt 2 GL3(Qp),λ L(λ), Σ(λ, L 1 , L 2 ) ∼ -→ Ext 2 GL3(Qp),λ L(λ), Σ + (λ, L 1 , L 2 )which together with Lemma 4.2.35 implies that Ext 2 GL3(Qp),λ L(λ), Σ + (λ, L 1 , L 2 ) admits a basis of the form {κ(b 1,valp ∧ b 2,valp ), ι 1 (D 0 )}, and therefore the element ι 1 (D 0 ) + L κ(b 1,valp ∧ b 2,valp )

  ),λ L(λ), Σ ,+, (λ,L ) → Ext 1 GL3(Qp),λ L(λ), L(λ) ⊗ E v ∞ P2 ⊕ v ∞ P1 (4.6.19)by Lemma 4.6.14. Note that we havedim E Ext 1 GL3(Qp),λ L(λ), L(λ) ⊗ E v ∞ P2 ⊕ v ∞ P1 = 2uniquely factors through a compositionΣ ,+ (λ, L 1 , L 2 ) [-1] → L(λ) ⊗ E v ∞ P3-i → L(λ) [1]which induces a commutative diagram with four distinguished trianglesΣ ,+ (λ, L 1 , L 2 ) [-1] L(λ) ⊗ E v ∞ P3-i L(λ) [1] Σ ,+ i (λ, L 1 , L 2 , L 3 ) D(λ, L 1 , L 2 , L 3 ) Hence we deduce that Σ ,+ i (λ, L 1 , L 2 , L 3 ) -→ D(λ, L 1 , L 2 , L 3 ) -→ W 3-i [1] +1 --→ or equivalently W 3-i -→ Σ ,+ i (λ, L 1 , L 2 , L 3 ) -→ D(λ, L 1 , L 2 , L 3 ) +1 --→

  Remark 4.6.58. It is actually possible to prove (by several technical computations of Ext-groups) that the quotientΣ min,+ (λ, L 1 , L 2 , L 3 )/L(λ) ⊗ E St ∞ 3 and the quotient Σ min (λ, L 1 , L 2 , L 3 )/L(λ) ⊗ E St ∞ 3 are independent of the choices of L 1 , L 2 , L 3 ∈ E.

  7.10) by reducing to the isomorphismHom GL3(Qp) Fil k+1 Σ max (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] ∼ -→ Hom GL3(Qp) Fil k Σ max (λ, L 1 , L 2 , L 3 ) ⊗ E (ur(α) ⊗ E ε 2 ) • det, S(U p , E) an [m ρ ] (4.7.12) for each k ∈ Z. If Gr k := Fil k+1 Σ min (λ, L 1 , L 2 , L 3 )/Fil k Σ min (λ, L 1 , L 2 , L 3 )

  Integral p-adic Hodge theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Breuil modules with descent data . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Linear algebra with descent data . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Fontaine-Laffaille modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Étale φ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Local Galois side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Elimination of Galois types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Fontaine-Laffaille parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Breuil modules of certain inertial types of niveau 1 . . . . . . . . . . . . . . . . 3.3.4 Fontaine-Laffaille parameters vs Frobenius eigenvalues . . . . . . . . . . . . . . 3.3.5 Filtration of strongly divisible modules . . . . . . . . . . . . . . . . . . . . . . . 3.3.6 Reducibility of certain lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Main results in characteristic p . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.4.3 Summary of results on Deligne-Lusztig representations . . . . . . . . . . . . . 86 3.4.4 Proof of Theorem 3.4.36 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.4.5 Jacobi sums in characteristic 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.4.6 Special vectors in a dual Weyl module . . . . . . . . . . . . . . . . . . . . . . . 96 3.4.7 Some technical formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 3.4.8 Proof of Theorem 3.4.35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 3.5 Mod p local-global compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 3.5.1 The space of algebraic automorphic forms . . . . . . . . . . . . . . . . . . . . . 120 3.5.2 Serre weights and potentially crystalline lifts . . . . . . . . . . . . . . . . . . . 122 3.5.3 Weight elimination and automorphy of a Serre weight . . . . . . . . . . . . . . 125 3.5.4 Some application of Morita theory . . . . . . . . . . . . . . . . . . . . . . . . . 126 3.5.5 Generalization of Section 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.5.6 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 First of all, I would like to thank my supervisor Prof. Christophe Breuil for guiding me into this extremely rich area of p-adic local Langlands correspondence and p-adic local-global compatibility. I have tremendously benefited from discussions with him and the two thesis problems suggested by him are perfect first-step training for me towards my future academic career. I very much thank him for his really careful corrections on an early draft of my thesis and for his help on improving my writing skills. My opinions towards mathematical research has been deeply influenced by him throughout my Ph.D. I would like to thank Prof. David Savitt and Prof. Tobias Schmidt for kindly accepting to write reports for this thesis. I would like to thank Prof. Laurent Berger, Prof. Jean-François Dat, Prof. Stefano Morra, Prof. Benjamin Schraen for being part of the members of the committee.

1 Introduction générale 1.1 Conjecture de compatibilité locale-globale p-adique . . . . . . . . . . . . . . . . . . . . 1.2 Compatibilité local-global modulo p pour GL n (Q p ) . . . . . . . . . . . . . . . . . . . . 1.3 Dilogarithme et L -invariants supérieurs pour GL 3 (Q p ) . . . . . . . . . . . . . . . . . 2 General Introduction 2.1 p-adic local-global compatibility conjecture . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mod p local-global compatibility for GL n (Q p ) . . . . . . . . . . . . . . . . . . . . . . . 2.3 Dilogarithm and higher L -invariants for GL 3 (Q p ) . . . . . . . . . . . . . . . . . . . . 3 Mod p local-global compatibility for GL n (Q p ) in the ordinary case 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Local Galois side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Local automorphic side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Weight elimination and automorphy of a Serre weight . . . . . . . . . . . . . . 3.1.4 Mod p local-global compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 3.2.1 Filtered (φ, N )-modules with descent data . . . . . . . . . . . . . . . . . . . . . 3.2.2 Strongly divisible modules with descent data . . . . . . . . . . . . . . . . . . . 3.2.3 3.3.7 Main results on the Galois side . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Local automorphic side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Jacobi sums in characteristic p . . . . . . . . . . . . . . . . . . . . . . . . . . . i CONTENTS 3.4.2

  so that we can now apply the conditions in (3.3.16) as well as Definition 3.3.18 to the Breuil 60CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE modules in Lemma 3.3.21. Moreover, H can be written as

  5. MOD P LOCAL-GLOBAL COMPATIBILITY 123 which is a Serre weight for G n (O F + ,p ). From [EGH13], Lemma 7.3.4 if V is a Serre weight for G n , there exists a restricted weight a = (a w ) w

  ) for the abelian category consisting of admissible locally Q p -analytic representations of H (resp. admissible smooth representations of H) with coefficient E. Therefore taking strong dual induces a faithful contravariant functor from Rep la H,E to the abelian category Mod D(H,E) of abstract modules over D(H, E). The E-vector space Ext i D(H,E) (M 1 , M 2 ) is well-defined for any two objects M 1 , M 2 ∈ Mod D(H,E) , and therefore we define , Π 2 ∈ Rep la H,E where • is the notation for strong dual. We also define the cohomology of an object M ∈ Mod D(H,E) by H is the strong dual of the trivial representation of H. If H is a closed locally Q p -analytic normal subgroup of H, then H/H is also a locally Q p -analytic group. It follows from the fact D(H, E) ⊗ D(H ,E) E ∼ = D(H/H , E) (see Section 5.1 of [Bre17] for example) that H i (H , M ) admits a structure of D(H/H , E)-module for each M ∈ Mod D(H,E) . We define the H -homology of Π ∈ Rep la H,E as the object (if it exists up to isomorphism) H i (H , Π) ∈ Rep la H/H ,E such thatH i (H , Π) ∼ = H i (H , Π ).We emphasize that H i (H , Π) is well defined in the sense above only after we know its existence.We fix a subgroup Z of the center of the group H, then the algebra D(Z, E) consisting of locally Q p -analytic distribution with coefficient E on Z is naturally contained in the center of D(H, E). For each locally Q p -analytic E-character χ of Z, we can define the abelian subcategory Mod D(H,E),χ consisting of all the objects in Mod D(H,E) on which D(Z, E) acts by χ . Then we consider the functors Ext i D(H,E) (-, -) defined as Ext i Mod D(H,E),χ (-, -) which are extensions inside the abelian category Mod D(H,E),χ . Consequently we can define Ext i H,χ (Π 1 , Π 2 ) := Ext i D(H,E),χ (Π 2 , Π 1 ) for any two objects Π 1 , Π 2 ∈ Rep la H,E such that Π 1 , Π 2 ∈ Mod D(H,E),χ . In particular, if Z is the center of H and acts on Π ∈ Rep la H,E via the character χ, then Π ∈ Mod D(H,E),χ , and we usually say that Π admits a central character χ. Assume now H is the set of Q p -points of a split reductive group over Q p . We recall the category O together with its subcategory O p alg for each parabolic subgroup P ⊆ H from Section 9.3 of [Hum08] or [OS15]. The construction by Orlik-Strauch in [OS15] gives us a functor We use the notation Rep OS H,E for the abelian full subcategory of Rep la

	(resp. Rep ∞ H,E Ext i H (Π 1 , Π 2 ) := Ext i D(H,E) (Π 2 , Π 1 ) for any two objects Π 1 F H	la H,E

i (H, M ) := Ext i D(H,E) (1, M ) where 1 P : O p alg × Rep ∞ L,E → Rep la H,E

for each parabolic subgroup P ⊆ H with Levi quotient L.

  4.2.1. We have a spectral sequeceExt j L, * (H k (N, Π 1 ), Π 2 ) ⇒ Ext j+k H, * Π 1 , Ind H P (Π 2 ) an . Hom H, * Π 1 , Ind H P (Π 2 ) Ext 2 L, * (H 0 (N, Π 1 ), Π 2 ) for each Π 1 ∈ Rep la H,E , Π 2 ∈ Rep laL,E satisfying the (FIN) condition in Section 6 of[START_REF] Schneider | Duality for admissible locally analytic representations[END_REF], * ∈ {∅, χ} where χ is a locally analytic character of the center of H.

	which implies an isomorphism
	Hom L, * (H 0 (N, Π 1 ), Π 2 )

∼

-→ an and a long exact sequence

Ext 1 L, * (H 0 (N, Π 1 ), Π 2 ) → Ext 1 H, * Π 1 , Ind H P (Π 2 ) an → Hom L, * (H 1 (N, Π 1 ), Π 2 ) →

  constructed an irreducible locally analytic representation Lemma 4.2.2. The functor -⊗ E δ induces an equivalence of category from Rep la H,E to itself. Moreover, the restriction of -⊗ E δ to the subcategory Rep OS H,E is again an equivalence of category to itself and satisfies

	F H P (M, π ∞ )
	of H.

  Remark 4.2.12. It is actually possible to show that Ω is the set of (isomorphism classes of ) irreducible objects of the block inside Rep OS GL3(Qp),E containing the object L(λ).

				(4.2.11)
			C 1 s2s1,s2s1	C 2 s2s1,s2s1
	C s1,s1	C s1s2,s1	C s2,s2	C s2s1,s2
	C s1s2s1,w	w ∈ W

Lemma 4.2.13. The representation v an Pi (λ) fits into a non-split extension

L(λ) ⊗ E v ∞ Pi → v an Pi (λ) C 1 s3-i,1

(4.2.14)

  that Σ GL2 (ν, L ) has the form {Σ GL2 (ν, L ) | L ∈ E} exhausts (up to isomorphism) all different locally analytic Erepresentations of GL 2 (Q p ) of the form (4.2.22) that do not contain L GL2 (ν) ⊗ E St ∞ : GL 2 → L i for i = 1,2 by identifying GL 2 with a Levi block of L i , which induce the embeddings ι T,i : T 2 → T and we have Σ(λ, L 1 , L 2 ) ∼ = Σ(λ, L , L ) (4.2.27) if L 1 = -L , L 2 = -L ∈ E, (4.2.28)

	St an 2 (ν)	L GL2 (ν)	(4.2.22)
	and the set 2	L GL2 (ν)
	as a subrepresentation. We have the embeddings
	ι

i

  Li , St an 2 )We use the notation M H (λ H ) µ for the subspace of M H (λ) with t h -weight µ and note thatdim E M H (λ H ) λ H = 1.We easily observe thatZ(U (h)) • M H (λ H ) λ H = M H (λ H ) λ H and U (t h ) • M H (λ H ) λ H = M H (λ H ) λ H . (4.3.5)It is well-known that the the direct sum decompositionh = n H ⊕ t h ⊕ n H (4.3.6)induces a tensor decomposition of E-vector spaceU (h) = U (n H ) ⊗ E U (t h ) ⊗ E U (n H ). (4.3.7)Hence we can write each element in U (h) as a polynomial with variables indexed by a standard basis of h that is compatible with (4.3.6). It follows from the definition of θ as the restriction to Z(U (h)) of the projection U (h) U (t h ) (coming from (4.3.7)) thatj 1 (z) -j 2 • θ(z) ∈ U (h) • n H + n H • U (h)for each z ∈ Z(U (h)). If a monomial f in the decomposition (4.3.7) of j 1 (z) -j 2 • θ(z) belongs to n

		2 GL2(Qp),0 (1, St an 2 )
	(c.f. (5.38) of [Schr11]) which naturally embeds into Ext 2 GL2(Qp) (1, St an 2 ). Then the map ι i : GL 2 →
	L i induces the isomorphisms
	Ext 2 GL2(Qp) (1 2 , St an 2 )	∼ ← -Ext 2 GL3(Qp),0 1 3 , I GL3 Pi (St an 2 )	(4.2.31)

∼ ← -Ext 2 Li(Qp),0 (1 H • U (n H ) • U (t h ), then we have f • M H (λ H ) λ H ⊆ M H (λ H ) µ

  ,1 C si,si (4.4.35) is split or not. However, If (4.4.35) is split, then there exists a representation of the form

	C 2 si,1

From another point of view, to a admissible smooth F p -representation Π of GL n (K) for a finite extension K of Q p , Scholze[START_REF] Scholze | On the p-adic cohomology of the Lubin-Tate tower[END_REF] attaches a admissible smooth F p -representation S(Π) of D × for

copies of the projective line P 1 (F),

Acknowledgement Acknowledgements

The authors express their deepest gratitude to Christophe Breuil for his encouragements, his careful reading of an earlier draft of this paper, and his numerous helpful comments and suggestions. The authors also sincerely thank Florian Herzig for his constant interest, his many helpful comments and suggestions, and pointing out some mistakes in an earlier draft of this paper. The authors thank Yiwen Ding, Yongquan Hu, Bao V. Le Hung, Daniel Le and Stefano Morra for plenty of discussions. The second named author thanks James E. Humphreys for kind guidance through messages on references for modular representation theory. The second named author thanks Jian Qian for giving some numerical evidence through computer programming. The first named author thanks Seunghwan Chang for his helpful comments and suggestions.

Theorem 3.4.39 ([Jan81], Satz 4.3). Assume that µ + η is in the lowest restricted alcove and 2ngeneric (Definition 3.4.5). Then we have

[ Z 1 (µ -pν + pη) : L 1 ( w • µ)]F ( w • (µ + wν)).

Proposition 3.4.40. Let λ ∈ X(T ) + . Suppose µ ∈ X(T ) is maximal for µ ↑ λ and µ = λ. If µ ∈ X(T ) + and if µ = λ -pα for all α ∈ Φ + , then

Proof. This is the Corollary II 6.24 in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF].

If M is an arbitrary G-module, we use the notation M [1] for the Frobenius twist of M as defined in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF], I.9.10. Proposition 3.4.41 ([Jan03], Proposition II. 9.14). Let λ ∈ X(T ) + . Suppose each composition factor of Z 1 (λ) has the form L 1 (µ 0 + pµ 1 ) with µ 0 ∈ X 1 (T ) and µ 1 ∈ X(T ) such that µ 1 + η, β ∨ ≥ 0 for all β ∈ ∆. Then H 0 (λ) has a filtration with factors of the form F (µ 0 ) ⊗ H 0 (µ 1 ) [1] . Each such module occurs as often as L 1 (µ 0 + pµ 1 ) occurs in a composition series of Z 1 (λ).

Remark 3.4.42. Note that if µ 1 is in the lowest restricted alcove, then F (µ 0 ) ⊗ H 0 (µ 1 ) [1] = F (µ).

Lemma 3.4.43 ([Jan03], Lemma II. 9.18 (a)). Let L 1 (µ) be a composition factor of Z 1 (λ), and write λ + η = pλ 1 + λ 0 and µ = pµ 1 + µ 0 with λ 0 , µ 0 ∈ X 1 (T ) and λ 1 , µ 1 ∈ X(T ).

If λ, α ∨ ≥ n -2 (3.4.44) for all α ∈ Φ + , then µ 1 + η, β ∨ ≥ 0 for all β ∈ Φ + .

Proof. We only need to mention that h α = n for all α ∈ Φ + and for our group G = GL n/Fp , where h α is defined in [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF], II.9.18.

We define an element s α,m ∈ W by

for each α ∈ Φ + and m ∈ Z.

Theorem 3.4.45. Let λ, µ ∈ X(T ) such that µ = s α,m • λ and mp < λ + η, α ∨ < (m + 1)p.

(3.4.46)

Assume further that there exists ν ∈ X(T ) such that λ + pν satisfies the condition (3.4.44) and that ν and µ 1 + ν are in the lowest restricted alcove. Then we have [ Z 1 (λ) : L 1 (µ)] = 1.

Lemma 3.4.127. For 1 ≤ r ≤ n -1, we have the following equalities on H 0 (µ w0 0 ) µ * :

for all 1 ≤ k ≤ p -1.

Proof. Note that we have

Therefore µ w0 0 -(µ * + kβ) / ∈ α∈Φ + Z ≥0 α as long as k > [a 0 -a n-1 ] 1 + n -2. As (a n-1 , • • • , a 0 ) is assumed to be n-generic in the lowest alcove throughout this section, we deduce that

On the other hand, by the definition (3.4.1), the image of X alg β,k lies inside H 0 (µ w0 0 ) µ * +kβ , which is zero by (3.4.128) assuming k ≥ p -1. Hence we deduce that X alg β,k = 0 on H 0 (µ w0 0 ) µ * for all k ≥ p -1.

Then the conclusion of this lemma follows from the equality (3.4.10).

We have a natural embedding H 0 (µ w0 0 ) → π 0 by the definition of algebraic induction and parabolic induction. Recall that we have defined U 1 in Example 3.4.23.

Lemma 3.4.129. We have F p [S k ,w0 v 0 ] = H 0 (µ w0 0 ) U 1 µ * . In particular,

V ⊆ H 0 (µ w0 0 ).

Proof. On one hand, by Corollary 3.4.80 we know that dim Fp H 0 (µ w0 0 ) U 1 µ * = 1, and this space is generated by v alg {n-1},m where

(3.4.130)

We now need to identify the vector v alg {n-1},m with certain linear combination of Jacobi sums. Note that by Corollary 3.4.80 we have

an-i-an-i-1 i .

Given a matrix A ∈ G(F p ), then D i (A) = 0 for all 1 ≤ i ≤ n -1 if and only if

and thus the support of v alg {n-1},m is contained in B(F p )w 0 B(F p ). As a result, by the proof of Proposition 3.4.18, we know that v alg {n-1},m is a linear combination of vectors of the form S k,w0 v 0 .

114CHAPTER 3. MOD P LOCAL-GLOBAL COMPATIBILITY FOR GL N (Q P ) IN THE ORDINARY CASE As v alg {n-1},m is U 1 -invariant, and in particular U 1 (F p )-invariant, then by Proposition 3.4.30 we know that it is a linear combination of vectors of the form S k,w0 v 0 (3.4.131) such that k 1,n = [a 0 -a n-1 ] 1 + n -2, k 1,j = 0 or p -1 for 2 ≤ j ≤ n -1 and k i,j = 0 for all 2 ≤ i < j ≤ n.

Finally, note that

an-i-an-i-1 i is a polynomial of t with degree [a 0 -a n-1 ] + n -2, we conclude that

where 0 is the (n -1)-tuple with all entries zero. By Lemma 3.4.127 and the fact that

we deduce that X β,[a0-an-1]1+n-2 v alg {n-1},m = c S 0,w0 v 0 for some non-zero constant c . By Lemma 3.4.126 and the linear independence of Jacobi sums proved in Proposition 3.4.18, we know that only S k ,w0 v 0 can appear in the linear combination 3.4.131. In other words, we have shown that v alg {n-1},m = c S k ,w0 v 0 for some non-zero constant c , and thus we finish the proof.

Lemma 3.4.132. The dual Weyl module H 0 (µ w0 0 ) is uniserial with length two with socle F (µ w0 0 ) and cosocle F (µ * ).

Proof. By [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] Proposition II.2.2 we know that soc G H 0 (µ w0 0 ) is irreducible and can be identified with F (µ w0 0 ) (which is in fact the definition of F (µ w0 0 )). Therefore it suffices to show that H 0 (µ w0 0 ) has only two Jordan-Hölder factor F (µ w0 0 ) and F (µ * ), each of which has multiplicity one. By [Jan03] II.2.13 (2) it is harmless for us to replace H 0 (µ w0 0 ) by the Weyl module V (µ w0 0 ) (defined in [Jan03] II.2.13) and show that V (µ w0 0 ) has only two Jordan-Hölder factor F (µ w0 0 ) and F (µ * ) and each of them has multiplicity one. As

we deduce that the only dominant alcove lying below the one µ w0 0 lies in is the lowest p-restricted alcove. In particular, the only dominant weight which is linked to and strictly smaller than µ w0 0 is µ * . By [START_REF] Jantzen | Representation of Algebraic Groups Second Edition[END_REF] Proposition II. 8.19, we know the existence of a filtration of subrepresentation

• For each place w|p of F , r Π | G Fw is potentially diagonalizable;

• r(G F (ζp) ) is adequate.

If a = (a w ) w ∈ (Z n + )

Sp 0 and for each w ∈ S p r| G Fw has a potentially diagonalizable crystalline lift with Hodge-Tate weights {a 1,w + (n -1), a 2,w + (n -2), • • • , a n-1,w + 1, a n,w }, then a Jordan-Hölder factor of W a ⊗ Zp F is a Serre weight of r.

Weight elimination and automorphy of a Serre weight

In this section, we state our main Conjecture for weight elimination (Conjecture 3.5.16) which will be a crucial assumption in the proof of Theorem 3.5.44. This conjecture is now known by Bao V. Le Hung (c.f. [LLMPQ]). We also prove the automorphy of a certain obvious Serre weight under the assumptions of Taylor-Wiles type.

Throughout this section, we assume that ρ 0 is always a restriction of an automorphic representation r : G F → GL n (F) to G Fw for a fixed place w above p and is generic (c.f. Definition 3.3.3). Recall that for 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1 we have defined a tuple of integers (r i0,j0 n-1 , • • • , r i0,j0 1 , r i0,j0 0

) in (3.3.42), which determines the Galois types as in (3.1.3). In many cases, we will consider the dual of our Serre weights, so that we define a pair of integers (i 1 , j 1 ) by the equation (3.5.3). We also let b k := -c n-1-k for all 0 ≤ k ≤ n -1. We will keep the notation (i 1 , j 1 ) and b k for the rest of the paper.

For the rest of the this section, we are mainly interested in the following characters of T (F p ): let

and µ ,i1,j1 := (y n-1 , y n-2 , • • • , y 1 , y 0 )

where

As ρ 0 is generic, each of the characters above is p-regular and thus uniquely determines a p-restricted weight up to a twist in (p -1)X 0 (T ), and, by abuse of notation, we write µ , µ ,i1,j1 for those corresponding p-restricted weights, respectively. We will clarify the twist in (p -1)X 0 (T ) whenever necessary. We also define a principal series representation

(3.5.15)

We now state necessary results of weight elimination to our proof of the main results, Theorem 3.5.44, in this paper.

Conjecture 3.5.16. Let r : G F → GL n (F) be a continuous automorphic Galois representation with r| G Fw ∼ = ρ 0 as in (3.3.1). Fix a pair of integers (i 0 , j 0 ) such that 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1, and assume that ρ i0,j0 is Fontaine-Laffaille generic and that µ ,i1,j1 is 2n-generic.

Then we have

In an earlier version of this paper, we prove Conjecture 3.1.11 for n ≤ 5. But our method is rather elaborate to execute for general n. We are informed that Bao V. Le Hung can prove Conjecture 3.1.11 completely. Therefore, Conjecture 3.1.11 becomes a theorem based on the results in the forthcoming paper [LLMPQ].

Finally, we prove the automorphy of the Serre weight F (µ ) ∨ .

Proposition 3.5.17. Keep the assumptions and notation of Conjecture 3.5.16. Assume further that if n is even then so is n

) is an irreducible representation with split ramification, and that there is a RACSDC automorphic representation Π of GL n (A F ) such that

Proof. We prove that F (µ

is in the lowest alcove as ρ 0 is generic, so that by Theorem 3.5.14 it is enough to show that ρ 0 has a potentially diagonalizable crystalline lift with Hodge-Tate weights [BLGGT], Lemma 1.4.3 it is enough to show that ρ 0 has an ordinary crystalline lift with those Hodge-Tate weights. The existence of such a crystalline lift is immediate by [GHLS], Proposition 2.1.10. On the other hand, we have

) ∨ which is a direct corollary of Theorem 3.5.28. Therefore, we conclude that

Some application of Morita theory

In this section, we will recall standard results from Morita theory to prove Corollary 3.5.23. We fix here an arbitrary finite group H and a finite dimensional irreducible E-representation V of H. We may assume that E is sufficiently large such that E (resp. its residual field F) is a splitting field of V . By Proposition 16.16 in [START_REF] Curtis | Methods of Representation Theory: With Applications to Finite Groups and Orders[END_REF], we know that for any

depends only on V and is independent of the choice of V

By considering an O E -lattice in the E-dual of V with the F-dual of σ as cosocle and then taking O E -dual of this lattice, we reach another O E -lattice V σ in V , which is the unique (up to homethety), such that soc

By repeating the proof of Lemma 2.3.1, Lemma 2.3.2 and Proposition 2.3.3 in [START_REF] Le | Lattices in the cohomology of U (3) arithematic manifolds[END_REF], we deduce the following. Proposition 3.5.18. If σ has multiplicity one in V , then the lattice V σ is a projective object in C.

We need to emphasize that the proof of Proposition 2.3.3 in [START_REF] Le | Lattices in the cohomology of U (3) arithematic manifolds[END_REF] requires only the multiplicity one of σ, although it is necessary for all Jordan-Hölder factors σ to have multiplicity one to have Proposition 2.3.4 in [START_REF] Le | Lattices in the cohomology of U (3) arithematic manifolds[END_REF].

then we have a surjection By Proposition 3.5.18 we know that σ∈Σ V σ is a projective object in C. By the definition of V σ we know that there is a surjection

which can be lifted by projectiveness to (3.5.21).

Note in particular that (3.5.21) implies automatically the surjection

Proof. This is simply the F-dual of (3.5.22).

Generalization of Section 3.4

In this section, we fix a pair of integers (i 0 , j 0 ) satisfying 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1, and determine (i 1 , j 1 ) by the equation (3.5.3). We will use the shorten notation P (resp. N , L, P -• • • ) for P i1,j1 (resp. N i1,j1 , L i1,j1 , P - i1,j1 , • • • ) as introduced at the beginning of Section 3.5. Proposition 3.5.38 is crucial for the proof of Theorem 3.5.44. We assume throughout this section that µ ,i1,j1 is 2n-generic (c.f. Definition 3.4.5).

We start this section by defining some weights and Jacobi sum operators which will play a crucial role for our main results, Theorem 3.5.44. Let

where

We also fix certain two elements in the Weyl group W :

and further define two more weights

More precisely, µ i1,j1 and µ i1,j1, can be written as follow:

where

Note that if we let

Recall that w L 0 is defined at the beginning of Section 3.5 and that µ ,i1,j1 is defined in Section 3.5.3. We now define certain mod p Jacobi sum operators:

Proof. The equality (3.5.31) follows directly from the proof of (ii) of Lemma 3.5.29. We define a new tuple k i1,j1,0 = (k i1,j1,0 i,j

We also define V i1,j1, to be the subrepresentation of π i1,j1 * generated by

) U (Fp),(µ ,i 1 ,j 1 ) w 0 .

By Proposition 3.4.101 and the same method in the proof of Proposition 3.4.124 we deduce that

By abuse of notation we view µ ,i1,j1 as a fixed weight in X 1 (T ), and then there exists µ , ∈ X + (T ) such that

We define U i1,j1 1 to be the unipotent subgroup of L generated by U αr for n -

By a direct generalization of proof of Lemma 3.4.129, we can show that

We define V i1,j1 alg to be the G-subrepresentation of H 0 (µ ,i1,j1 ) generated by H 0 (µ ,i1,j1

and by definition we have

(3.5.35)

We have natural identification (c.f. the beginning of Section 3.5 for definition of H 0 L (µ ,i1,j1 ))

) and H 0 (µ ,i1,j1

(3.5.36) By applying Lemma 3.4.132 and the proof of Proposition 3.4.133 to the Levi L, we deduce that H 0 L (µ ,i1,j1 ) is uniserial of length two with socle F L (µ ,i1,j1 ) and cosocle F L (µ , ) and that

and thus a non-zero morphism

by coinduction for algebraic representation from P to G. In particular we know that

alg . Now we restrict the action of G to G(F p ) and observe the injections

which induces

Hence we deduce that

which together with (3.5.34) finishes the proof of (3.5.32).

Proposition 3.5.38. Let τ be an

(ii) We have the non-vanishing results for S i1,j1 and S i1,j1, :

(iii) We also have the non-vanishing results for S i1,j1 1 and S i1,j1, 1 :

Proof. We can easily deduce (i) from

and Frobenius reciprocity as F (µ i1,j1 ), F (µ i1,j1 ), F (µ i1,j1 ) and F (µ i1,j1 ) all have multiplicity one in τ ⊗ O E F. We define π i1,j1 as the mod p reduction of ( π i1,j1 *

Then we deduce from Corollary 3.5.23 that there exists an injection

Note that we have

}. The equality of two spaces in (ii) is true because both of them can be identified with S k i 1 ,j 1 ,0 ,w0 (τ ⊗ O E F) U (Fp),(µ ,i 1 ,j 1 ) w 0 by the same argument as in the proof of (ii) of Lemma 3.5.29. Therefore we only need to show that S i1,j1 (resp. S i1,j1, ) gives rise to a bijection from π i1,j1 * ⊕ π i1,j1 U (Fp),µ i 1 ,j 1 (resp. from

) to its image. According to (ii) of Lemma 3.5.29 and (3.5.39) we only need to show that S i1,j1 (π i1,j1 ) U (Fp),µ i 1 ,j 1 = 0 and S i1,j1, (π i1,j1 ) U (Fp),µ i 1 ,j 1 , = 0 which follows from Lemma 3.5.30 by definition of π i1,j1 . We have a unique (up to scalar) non-zero morphism

which by Lemma 3.5.30 induces isomorphisms

for µ ∈ {µ i1,j1 , µ i1,j1, }, and hence (iii) follows from (iii) of Lemma 3.5.29 by considering the image of (iii) of Lemma 3.5.29 under (3.5.40) inside π i1,j1 .

Corollary 3.5.41. Let τ be an

).

Then we have

Main results

In this section, we state and prove our main results on mod p local-global compatibility. Throughout this section, ρ 0 is always assumed to be a restriction of a global representation r : G F → GL n (F) to G Fw for a fixed place w of F above p. Let v := w| F + , and assume further that r is automorphic of a Serre weight

+ where w is a place of F above v , and define

(3.5.42)

From now on, we also assume that a w is in the lowest alcove for each place w of F above p, so that

), which is sufficiently small and unramified above p, such that S(U, V )[m r ] = 0 where m r is the maximal ideal of T P attached to r for a cofinite subset P of P U .

We fix a pair of integers (i 0 , j 0 ) such that 0 ≤ j 0 < j 0 + 1 < i 0 ≤ n -1, and determine a pair inters (i 1 , j 1 ) by the equation (3.5.3). We also define

We fix a finite length locally analytic representation V ∈ Rep la H,E equipped with a increasing filtration of subrepresentations {Fil k V } 0≤k≤m such that Fil 0 (V ) = 0, Fil m (V ) = V and gr k+1 V := Fil k+1 V /Fil k V = 0 for all 0 ≤ k ≤ m -1.

Note that the assumption above automatically implies that (V ) ≥ m where (V ) is the length of V . Proposition 4.2.5. Assume that W is another object of Rep la H,E and χ is a locally analytic character of the center of H.

Therefore we finish the proof of part (i) and the first claim of part (ii) by induction on k and the fact that gr 1 V = Fil 1 V . It remains to show the second claim of part (ii). The same method as in the proof of part (i

which finishes the proof by combining (4.2.7) and (4.2.8).

for i = 1, 2. On the other hand, the representation St an 3 (λ) has the following form:

Proof. The non-split short exact sequence follows directly from (3.62) of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF]. It follows easily from the definition of St an 3 (λ) and the main theorem of [START_REF] Orlik | On Jordan-Hölder series of some locally analytic principal series representations[END_REF] that

and each Jordan-Hölder factor occurs with multiplicity one. It follows from Section 5.2 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] that

for i = 1, 2. We also observe from Section 5.2 and 5.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] that

which together with

(4.2.17)

are subquotients of St an 3 (λ) by various properties of the functors F GL3 Pi (c.f. main theorem of [START_REF] Orlik | On Jordan-Hölder series of some locally analytic principal series representations[END_REF]) and the definition of St an 3 (λ). We finish the proof by combining (4.2.16) and (4.2.17) with the results before Remark 3.38 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF].

Remark 4.2.18. It is actually possible to show that all the possibly non-split extensions indicated in (4.2.15) are non-split, although they are essentially unrelated to the p-adic dilogarithm function.

p-adic logarithm and dilogarithm

In this section, we recall p-adic logarithm and dilogarithm function as well as their representation theoretic interpretations.

We recall the p-adic logarithm function log 0 : Q × p → Q p defined by the power series

We use the notation ι T,i (V L ) for the locally analytic representation of

and is trivial after restricting to the other copy of Q × p . By a direct analogue of Σ GL2 (ν, L ), we can construct Σ Li (λ, L ) as the subrepresentation of

In fact, if we have λ| T2,ι T ,i = ν, then we obviously know that Σ Li (λ, L )| GL2,ιi ∼ = Σ GL2 (ν, L ) where the notation (•)| * , means the restriction of • to * via the embedding . We observe that the parabolic induction I GL3

Pi (Σ Li (λ, L )) fits into the exact sequence

According to Proposition 5.6 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] for example, we know that

and thus we can define Σ i (λ, L ) as the unique quotient of

by b i,log 0 (resp. by b i,valp ). We use the notation 1 T for the trivial character of T (Q p ). We use the same notation b i,log 0 and b i,valp for the image of log 0 and val p respectively under the embedding

induced by the maps

where p i is the section of ι T,i which is compatible with the projection L i GL 2 . Recall the elements c i,log , c i,val ∈ Ext 1 T (Qp),0 (1 T , 1 T ) constructed after (5.24) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and observe that

We notice that there exists canonical surjections

with kernel spanned by {c i,log , c i,val }, according to (5.70) and (5.71) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF]. Therefore the relation (4.2.24) reduces via the surjection (4.2.25) to

where L i (Q p ) acts on St an 2 via the projection p i . We abuse the notation for the composition

given by (4.2.31) and the surjection

by (5.20) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF].

Lemma 4.2.33. We have

and the set

Proof. This follows directly from (5.57) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and (4.2.24).

Lemma 4.2.34. There exists γ ∈ E × such that

Proof. This follows directly from Lemma 5.8 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and (4.2.24) if we take

where α ∈ E × is the constant in the statement of Lemma 5.8 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF].

Lemma 4.2.35. We have

Moreover, the image of

Proof. This follows directly from Corollary 5.17 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and (4.2.24).

We recall from (5.55) of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] that

where α is defined in Lemma 5.8 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF].

Lemma 4.2.37. Assume that L 3 ∈ E satisfies the equality

Then we have

Proof. All the equalities in this lemma are understood to be inside

without causing ambiguity. It follows from our assumption (4.2.38) that

which together with (4.2.36) imply that

from the proof of Corollary 5.17 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] and that Remark 4.2.42. We emphasize that we do not know whether

or not, which is of independent interest.

A key result for GL

In this section, we are going to prove Proposition 4.3.14 which will be a crucial ingredient for the proof of Lemma 4.5.8 and Proposition 4.6.8. We usually identify GL 2 (Q p ) with a Levi factor of a maximal parabolic of GL 3 when we apply the results from this section. We use the following shorten notation

for each weight ν ∈ X(T 2 ).

Lemma 4.3.1. We have

Proof. This is essentially contained in the proof of Theorem 3.14 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF]. In fact, we know that

= 1 which finish the proof by a simple devissage induced by the short exact sequence

We fix a split p-adic reductive group H and have a natural embedding

where D(H, E) {1} is the closed subalgebra of D(H, E) consisting of distributions supported at the identity element (c.f. [START_REF] Kohlhaase | Invariant distributions on p-adic analytic groups[END_REF]). The embedding above induces another embedding

by the main result of [START_REF] Kohlhaase | Invariant distributions on p-adic analytic groups[END_REF] where Z(•) is the notation for the center of a non-commutative algebra. We say that Π ∈ Rep la GL2(Qp),E has an infinitesimal character if Z(U (h)) acts on Π via a character.

have both the same central character and the same infinitesimal character and satisfy Hom H (V, W ) = 0, then any non-split extension of the form W V has both the same central character and the same infinitesimal character as the one for V and W .

Proof. This is a direct analogue of Lemma 3.1 in [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] and follows essentially from the fact that both D(Z(H), E) and Z(U (h)) are subalgebras of Z(D(H, E)) by [START_REF] Kohlhaase | Invariant distributions on p-adic analytic groups[END_REF].

We fix a Borel subgroup B H ⊆ H as well as its opposite Borel subgroup B H . We consider the split maximal torus T H := B H ∩ B H and use the notation N H (resp. N H ) for the unipotent radical of B H (resp. of B H ). We use the notation J B H (•) for Emertion's Jacquet functor.

H,E has an infinitesimal character, then U (t h ) W H (as a subalgebra of U (t h )) acts on J B H (V ) via a character where W H is the Weyl group of H.

Proof. We know by our assumption that Z(U (h)) acts on V (and hence on V as well) via a character. We note from (4.3.2) that Z(U (h)) commutes with D(N H , E) ⊆ D(H, E) and thus the action of

for the Harish-Chandra isomorphism (c.f. Section 1.7 of [START_REF] Humphreys | Representations of Semisimple Lie Algebras in the BGG Category O[END_REF]) and the notation j 1 and j 2 for the embeddings

We choose an arbitrary Verma module M H (λ H ) with highest weight λ H , namely we have

Proof. Assume that a representation V of the form (4.3.10) has an infinitesimal character. Note that V can be represented by an element in the space Ext

We consider the upper-triangular Borel subgroup B 2 and the diagonal split torus T 2 . Then by the proof of Lemma 3.20 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] we know that the Jacquet functor J B2 (c.f. [START_REF] Emerton | Jacquet modules of locally analytic representations of p-adic reductive groups I: constructions and first properties[END_REF] for the definition) induces a injection

It follows from Lemma 3.20 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] (up to changes on notation) that the image of the composition of (4.3.13) and (4.3.12) is a certain two dimensional subspace Ext 1 T2(Qp) (1, 1) L of Ext 1 T2(Qp) (1, 1) depending on L . More precisely, if we use the notation 1 , 2 for the two charaters

forms a basis of Ext 1 T2(Qp) (1, 1), and the subspace Ext 1 T2(Qp) (1, 1) L has a basis

It follows from Lemma 4.3.4 that U (t 2 ) WGL 2 acts on J B2 (V ) via a character where W GL2 is the notation for the Weyl group of GL 2 (Q p ). Therefore we deduce by a twisting that the the subspace of Ext 1 T2(Qp) (1, 1) corresponding to J B2 (V ) is killed by U (t 2 ) WGL 2 . We notice that the subspace M of Ext 1 T2(Qp) (1, 1) killed by U (t 2 ) WGL 2 is two dimensional with basis

and we have

. However, the representation given by the line

which is a contradiction. The proof of the second statement is a direct analogue as we observe that J B2 also induces the following embedding

We define Σ + 2 (ν, L ) as the unique (up to isomorphism) non-split extension of I(s•ν) by Σ GL2 (ν, L ) given by Lemma 4.3.1.

Computations of Ext I

In this section, we are going to compute various Ext-groups based on known results on group cohomology in Section 5.2 and 5.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF].

Proposition 4.4.1. The following spaces are one dimensional

in all the other cases where

Proof. This follows from a special case of Proposition 4.7 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] together with the main result of [START_REF] Orlik | On extensions of generalized Steinberg representations[END_REF].

Lemma 4.4.2. We have

as the other cases are similar. We observe that (4.4.3) is equivalent to the non-existence of a uniserial representation of the form

which is again equivalent to the vanishing

due to Proposition 4.4.1. The short exact sequence

. Therefore (4.4.5) follows from Lemma 4.2.1 and the facts that

On the other hand, the short exact sequence We define W 0 as the unique locally algebraic representation of length three satisfying

We also define the (unique up to isomorphism) locally algebraic representation of the form

for each i = 1, 2 Lemma 4.4.6. We have

Proof. The short exact sequence Recall that we have introduced a set Ω consisting of irreducible locally analytic representations of GL 3 (Q p ) in (4.2.11). We define the following subsets of Ω:

Proposition 4.4.7. We have an explicit formula for

Proof. This follows directly from Section 5.2 and 5.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF].

Lemma 4.4.8. For

Proof. We only prove the statements for V 0 = L(λ) as other cases are similar. If

then the conclusion follows from Proposition 4.4.1. If

for a smooth irreducible representation π ∞ j and j = 1 or 2, then it follows from Lemma 4.2.1 that

It follows from Proposition 4.4.7 and (4.4.9) that

We notice that Z(L j (Q p )) acts via different characters on L j (λ), L j (s 3-j •λ) and L j (s 3-j s j •λ)⊗ E π ∞ j , and thus we have the equalities

for each π ∞ j and j = 1, 2. If

for a smooth irreducible representation π ∞ j and j = 1 or 2, then the short exact sequence

which implies an isomorphism

for each smooth irreducible π ∞ j = 1 Lj , and therefore

Finally, similar methods together with Proposition 4.4.7 also show that Ext 1 GL3(Qp),λ L(λ),

for each w ∈ W .

We define

Then we define the following subsets of Ω -for i = 1, 2:

Proof. The proof is very similar to that of Lemma 4.4.15.

Lemma 4.4.16. For

Proof. We only prove the statements for V 0 = L(λ) as other cases are similar. If

then the conclusion follows from Proposition 4.4.1. We notice that Z(L j (Q p )) acts via different characters on L j (λ), L j (s 3-j • λ) and L j (s 3-j s j • λ) ⊗ E π ∞ j , and thus we have

On the other hand, we notice that

On the other hand, we have

The short exact sequence

which finishes the proof if

. Finally, similar methods together with Proposition 4.4.7 also show that

for each w ∈ W .

Lemma 4.4.27. We have

Proof. We only prove the first vanishing

as the other cases are similar. The embedding

It follows from Proposition 4.4.7 that

We notice that Z(L 3-i (Q p )) acts on L 3-i (λ) and L 3-i (s i • λ) via different characters and that Hom L3-i(Qp),λ i

Therefore we deduce from (4.4.30) the equalities

Hence we finish the proof of (4.4.28) by the embedding (4.4.29).

Lemma 4.4.31. We have for i = 1, 2:

Proof. We only prove that

as the other cases are similar. The surjection

and the embedding

It follows from Proposition 4.4.7 that

and

We notice that

)) (k = 0, 1) via a different character, and the only direct summand that produces the same character as

). However, we know that cosoc L3-i(Qp),λ I

As a result, we deduce the equalities

Hence we finish the proof of (4.4.32) by the embedding (4.4.33).

Lemma 4.4.34. There exists a unique representation of the form

Computations of Ext II

In this section, we are going to establish several computational results (most notably Lemma 4.5.8) which have crucial applications in Section 4.7. We firstly recall the definition of Σ i (λ, L ) for i = 1, 2 and L ∈ E right before (4.2.23).

Lemma 4.5.1. We have

Proof. We only prove that

as the other equality is similar. We note that Σ 1 (λ, L 1 ) admits a subrepresentation of the form

due to Lemma 3.34, Lemma 3.37 and Remark 3.38 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF]. Therefore Σ 1 (λ, L 1 )) admits a filtration such that W appears as one term of the filtration and the only reducible graded piece is

It follows from Lemma 4.4.1 and Proposition 4.2.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] as well as our Lemma 4.4.15 that Ext 1 GL3(Qp),λ (C s1,s1 , V ) = 0 (4.5.3) for all graded pieces V such that V = V 1 . On the other hand, we have Lemma 4.5.6. We have

Proof. By symmetry, it suffices to prove that

This follows immediately from Lemma 3.42 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] as our Σ + 1 (λ, L 1 ) can be identified with the locally analytic representation Π 1 (λ, ψ) defined before (3.76) of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] up to changes on notation.

We define Σ + 1 (λ, L 1 ) (resp. Σ + 2 (λ, L 2 )) as the unique non-split extension given by a non-zero element in Ext 1 GL3(Qp),λ (C s1,s1 , Σ 1 (λ, L 1 )) (resp. in Ext 1 GL3(Qp),λ (C s2,s2 , Σ 2 (λ, L 2 ))). Hence we may consider the amalgamate sum of Σ + 1 (λ, L 1 ) and Σ + 2 (λ, L 2 ) over St an 3 (λ) and denote it by Σ + (λ, L 1 , L 2 ). In particular, Σ + (λ, L 1 , L 2 ) has the following form St an 3 (λ)

.

Lemma 4.5.7. We have

Proof. The short exact sequence

induces the following long exact sequence

As a result, we can deduce

from Lemma 4.5.6 and the facts

by Proposition 4.4.1 and Lemma 4.4.8. The proof for

is similar.

Lemma 4.5.8. We have

Proof. We only need to show the vanishing

as the others are similar or easier. We define ν := λ T2,ι T ,1 (which is the restriction of λ from T to T 2 via the embedding ι T,1 : T 2 → T ) and view Σ + GL2 (ν, L 1 ) (which is defined before Proposition 4.3.14) as a locally analytic representation of L 1 (Q p ) via the projection p 1 : L 1 GL 2 and denote it by Σ + L1 (λ, L 1 ). We note by definition by of Σ 1 (λ, L 1 ) that we have an isomorphism

Therefore we can deduce from the short exact sequence

and the fact (up to viewing I(s • ν) as a locally analytic representation of L 1 (Q p ) via the projection p 1 )

that we have an injection

which induces an injection

where we use the shorten notation

Note that we have an exact sequence

It follows from Proposition 4.4.7 that

Therefore we observe that

according to Proposition 4.3.14 and the natural identification

As a result, we deduce

from Lemma 4.2.1. We know that Ext 2 GL3(Qp),λ W 2 , v an P2 (λ) L(λ) = 0 (4.5.12) due to Proposition 4.4.1, Lemma 4.4.16 and a simple devissage, and thus we finish the proof by (4.5.9), (4.5.10), (4.5.11) and (4.5.12).

Lemma 4.5.13. We have

Due to a similar argument using (4.5.17), we only need to show that

to finish the proof of (4.5.14). The short exact sequence to deduce (4.5.18) from (4.5.19). The short exact sequence

using Proposition 4.4.1 and Lemma 4.4.16. Therefore we only need to show that

The equality (4.5.24) follows from Lemma 4.2.1 and the facts

where the first equality essentially follows from Lemma 3.14 of [START_REF] Breuil | Higher L -invariant for GL 3 (Q p ) and local-global compatibility[END_REF] and the second equality follows from checking the action of Z(L i (Q p )). On the other hand, (4.5.23) follows from (4.5.20) and Proposition 4.4.1 by an easy devissage. Hence we finish the proof.

Proposition 4.5.25. The short exact sequence

induces the following isomorphisms

and

Proof. The vanishing from Lemma 4.5.8 implies that

is an injection and hence an isomorphism as both spaces have dimension three according to Lemma 4.5.6 and Lemma 4.5.13. The proof of (4.5.27) is similar. We emphasize that both (4.5.26) and (4.5.27) can be interpreted as the isomorphism given by the cup product with the one dimensional space

We define

Lemma 4.5.28. We have

Proof. We define Σ ,-(λ, L 1 , L 2 ) as the subrepresentation of Σ (λ, L 1 , L 2 ) that fits into the following short exact sequence

(4.5.30) It follows from Lemma 4.4.8 that

by part (i) of Proposition 4.2.5. On the other hand, we know from Lemma 4.4.8 and Lemma 4.4.27 that there is no uniserial representation of the form

for i = 1, 2. Hence we deduce from (4.5.30), (4.5.31), (4.5.32) and Proposition 4.2.5 that

Therefore (4.5.29) induces an injection

Assume first that (4.5.34) is a surjection, then we pick a representation W represented by a non-zero element in Ext 1 GL3(Qp),λ L(λ), Σ (λ, L 1 , L 2 ) lying in the preimage of Ext 1 GL3(Qp),λ L(λ), C 1 s2,1 under (4.5.34). We note that there is a short exact sequence

We observe that L(λ) lies above neither C 1 s1,1 nor L(λ) ⊗ E v ∞ P2 inside W by our definition and (4.5.32), and thus W is mapped to zero under the map

which means that W comes from an element in

and in particular Ext 1 GL3(Qp),λ L(λ), Σ 1 (λ, L 1 ) = 0 (4.5.35)

The short exact sequence

On the other hand, the short exact sequence

as we have

from Lemma 4.4.2. We combine Lemma 4.5.8, (4.5.36) and (4.5.38) and deduce that Ext 1 GL3(Qp),λ L(λ), Σ 1 (λ, L 1 ) = 0 which contradicts (4.5.35). In all, we have thus shown that

by combining Lemma 4.4.8. Finally, the vanishing

which finishes the proof by combining Lemma 4.2.35 and (4.5.39).

Lemma 4.5.40. We have

Proof. The short exact sequence

λ) = 0 from Proposition 4.4.1 and Lemma 4.4.8. We can actually observe from Lemma 4.4.8 that the only

for i = 1, 2. The short exact sequence

by (4.5.42). Finally, the short exact sequence (4.5.37) induces

from Lemma 4.4.6, and by Lemma 4.5.28 as well as (4.5.43).

Lemma 4.5.44. We have the inequality

Proof. We know that

for i, j = 1, 2 from Proposition 4.4.1 and Lemma 4.4.8, and thus

for i, j = 1, 2 which together with (4.5.20) imply that

by Lemma 4.4.8 and thus we have

where the last equality follows again from Lemma 4.4.8. We finish the proof by combining (4.5.45) and (4.5.46) with the inequality

Key exact sequences

Lemma 4.6.1. We have the inequality

Proof. The short exact sequence

induces the exact sequence

) = 1 + 1 = 2 by Lemma 4.4.8 and Lemma 4.4.16. We also know that dim E Ext 1 GL3(Qp),λ (W 0 , Σ(λ, L 1 , L 2 )) = 2 by Lemma 4.5.40, and thus we obtain the following inequality:

The short exact sequence

induces a long exact sequence

by (4.6.4) and Lemma 4.5.44. We observe that Σ + 1 (λ, L 1 ) admits a filtration whose only reducible graded piece is

and thus it follows from Lemma 4.4.8 and

for all graded pieces of such a filtration except the subrepresentation L(λ) ⊗ E St ∞ 3 . Hence we deduce by part (ii) of Proposition 4.2.5 an isomorphism of one dimensional spaces

Then the short exact sequence

) which together with (4.6.6) and (4.6.7) implies that dim E Ext 1 GL3(Qp),λ W 2 , Σ + 1 (λ, L 1 ) ≥ 1 which contradicts Lemma 4.5.8. Hence we finish the proof.

Proof. It follows from (4.5.17) that we only need to show that

These results follow from combining the long exact sequence

with Lemma 4.2.35 and the equalities

due to Proposition 4.4.1.

Lemma 4.6.14. We have

Proof. It follows from (4.5.17) that we only need to show the equalities

which follow from combining (4.6.12), Lemma 4.6.13 and the long exact sequence

We use the shorten notation L := (L 1 , L 2 , L 1 , L 2 ) for a tuple of four elements in E. We recall from Proposition 4.5.25 an isomorphism of two dimensional spaces

(4.6.16)

We emphasize that the isomorphism (4.6.16) can be naturally interpreted as the cup product map

and thus we have an embedding

for each i = 1, 2. We notice that the quotient Σ ,+, (λ, L 1 , L 2 )/Σ +, 1 (λ, L 1 ) fits into a short exact sequence

Hence it remains to show the equality

then there exists a uniserial representation of the form

which contradicts (4.6.24) and Lemma 4.4.27. As a result, we have shown that

which together with Proposition 4.4.1 and part (i) of Proposition 4.2.5 implies (4.6.21) and hence (4.6.23) as well concerning (4.6.22). Therefore we can combine (4.6.23) with Lemma 4.5.8 and conclude that Ext 1 GL3(Qp),λ W 2 , Σ ,+, (λ, L 1 , L 2 ) = 0 which contradicts (4.6.20). Consequently, the injection (4.6.19) must be strict and we finish the proof.

According to Lemma 4.6.14, the short exact sequence

,λ L(λ), Σ ,+ (λ, L 1 , L 2 ) (4.6.25) Proposition 4.6.26. We have dim E Ext 1 GL3(Qp),λ (L(λ), Σ ,+, (λ, L )) = 1 and the image of f is not contained in the image of the natural injection

We use the shorten notation for the two dimensional space

) . We actually have the following commutative diagram

where the middle vertical map is just an equality. We know that h is injective by the vanishing

and k has a one dimensional image by (4.6.15). Both i and j are injective due to (4.6.11) and (4.6.12). Therefore by a simple diagram chasing we have

by Lemma 4.6.14 and therefore dim E Ext 1 GL3(Qp),λ L(λ), Σ ,+, (λ, L ) = 1 by Lemma 4.6.18. Moreover, the map g has a one dimensional image and hence k • f has one dimensional image, meaning that the image of f has dimension one or two and is not contained in Ker(k), which is exactly the image of

by (4.6.15). In fact, the restriction of f to the direct summand Ext 1 GL3(Qp),λ L(λ), L(λ) ⊗ E v ∞ Pi is given by the cup product map with a non-zero element in the line of

via (4.6.16) by our definition of Σ ,+ (λ, L ) and it is obvious that ι 1 (D 0 ) + L i κ(b 1,valp ∧ b 2,valp ) does not lie in the image of (4.6.28) which is exactly the line Eκ(b 1,valp ∧ b 2,valp ).

Proposition 4.6.29. We have

Proof. It follows from (4.6.25) that Ext 1 GL3(Qp),λ L(λ), Σ ,+ (λ, L ) = 1 if and only if the image of f is one dimensional. Then we notice by the interpretation of f as cup product in Proposition 4.6.26 that the image of

for each i = 1, 2. Therefore the image of f is one dimensional if and only if the two lines for i = 1, 2 coincide which means that

We use the notation Σ ,+ (λ, L 1 , L 2 , L 3 ) for the representation Σ ,+ (λ, L ) when

We define Σ min (λ, L 1 , L 2 , L 3 ) as the unique representation (up to isomorphism) given by a non-zero element in Ext 1 GL3(Qp),λ L(λ), Σ ,+ (λ, L 1 , L 2 , L 3 ) according to Proposition 4.6.29. Therefore by our definition Σ min (λ, L 1 , L 2 , L 3 ) has the following form St an 3 (λ)

(4.6.30)

It follows from Proposition 4.4.1, Proposition 4.6.29, the definition of Σ min (λ, L 1 , L 2 , L 3 ) and an easy devissage that Ext 1 GL3(Qp),λ L(λ), Σ min (λ, L 1 , L 2 , L 3 ) = 0. (4.6.31)

Remark 4.6.32. The definition of the invariant L 3 ∈ E of Σ min (λ, L 1 , L 2 , L 3 ) obviously relies on the choice of a special p-adic dilogarithm function D 0 which is non-canonical. This is similar to the definition of the invariants L 1 , L 2 ∈ E which relies on the choice of a special p-adic logarithm function log 0 .

Lemma 4.6.33. We have

Moreover, if V is a locally analytic representation determined by a line

then there exists a unique L 3 ∈ E such that

Proof. The short exact sequence

together with Lemma 4.6.13 induce a commutative diagram

(4.6.34) where we use shorten notation

) and V ,+ for Σ ,+ (λ, L 1 , L 2 ) to save space. We observe that g 2 is an injection due to Lemma 4.6.13, k 1 is a surjection by the proof of Proposition 4.6.8, h 3 is an isomorphism by Proposition 4.4.1 and an easy devissage and finally h 2 is an injection. Assume that h 2 is not surjective, then any representation given by a non-zero element in Coker(h 2 ) admits a quotient of the form

V alg i (4.6.35) for i = 1 or 2 due to Lemma 4.4.8. However, it follows from Lemma 4.4.27 that there is no uniserial representation of the form (4.6.35), which implies that h 2 is indeed an isomorphism, and hence k 2 is surjective by a diagram chasing. Therefore we conclude that

The final claim on the existence of a unique L 3 follows from Proposition 4.6.29, our definition of Σ min (λ, L 1 , L 2 , L 3 ) and the observation that the restriction of k 2 to the direct summand

which can be interpreted as the cup product morphism with the one dimensional space Ext 1 GL3(Qp),λ L(λ), V alg i for i = 1, 2.

We define Σ ,+ i (λ, L 1 , L 2 , L 3 ) as the subrepresentation of Σ ,+ (λ, L 1 , L 2 , L 3 ) that fits into the short exact sequence

Pi

for each i = 1, 2. We use the notation D i (λ, L 1 , L 2 , L 3 ) for the object in the derived category D b Mod D(GL3(Qp),E) associated with the complex

Proposition 4.6.36. The object

fits into the distinguished triangle

for each i = 1, 2. Moreover, the element in

associated with the distinguished triangle (4.6.37) is

(4.6.39)

Proof. It follows from Proposition 3.2 of [START_REF] Schraen | Représentation localment analytiques de GL 3 (Q p )[END_REF] that there is a unique (up to isomorphism) object

that fits into a distinguished triangle

such that the element in Ext 2 GL3(Qp),λ L(λ), Σ(λ, L 1 , L 2 ) associated with (4.6.40) via (4.6.38) is (4.6.39). It follows from TR2 (c.f. Section 10.2.1 of [START_REF] Weibel | A introduction to homological algebra[END_REF]) that

is another distinguished triangle. The isomorphism (4.6.16) can be reinterpreted as the isomorphism

that fits into the short exact sequence

that fits into the short exact sequence

The short exact sequence (4.6.44) induces a long exact sequence

which easily implies that

by Proposition 4.4.1 and (4.6.31). On the other hand, we notice that Σ min,--(λ, L 1 , L 2 , L 3 ) admits a filtration whose only reducible graded piece is

Ext 1 GL3(Qp),λ L(λ), V = 0 for all graded pieces V of such a filtration by Lemma 4.4.8 and Lemma 4.4.27, which implies that Ext 1 GL3(Qp),λ L(λ), Σ min,--(λ, L 1 , L 2 , L 3 ) = 0.

Therefore (4.6.45) induces an injection of a two dimensional space into a four dimensional space

which allow us to identify

with a line in M min . We use the number 1, 2, 3, 4 to index the four representations

s2,1 and C 1 s1,1 respectively, and we use the notation M I for each subset I ⊆ {1, 2, 3, 4} to denote the corresponding subspace of M + with dimension the cardinality of I. For example, M {1,2} denotes the two dimensional subspace

Lemma 4.6.47. We have the following characterizations of M min inside M + :

It follows from (4.6.30) that

and thus M min ∩ M {3,4} is one dimensional which must coincide with M -. The proof of Lemma 4.6.1 implies that M ⊆ M {i,3,4} for i = 1, 2 and therefore M ∩ M {i,3,4} is one dimensional, which implies that

for i = 1, 2. We observe (c.f. Lemma 4.5.8) that

for each {i, j} = {3, 4}, {1, 2}. We define Σ min,-, (λ, L 1 , L 2 , L 3 ) as the unique subrepresentation of Σ min,-(λ, L 1 , L 2 , L 3 ) that fits into the short exact sequence

and then define

due to Proposition 4.4.1. We notice that we have a direct sum decomposition

where V i is a representation of the form

Switching V 1 and V 2 if necessary, we can assume by (4.6.48) that

On the other hand, we have an embedding

The short exact sequences

from Lemma 4.5.8 and (4.6.50). The surjection W 2 L(λ) induces an embedding

which together with (4.6.51) imply that Ext 1 GL3(Qp),λ L(λ), Σ 1 (λ, L 1 ) = 0 and hence Ext 1 GL3(Qp),λ L(λ), Σ +, 1 (λ, L 1 ) = 0 (4.6.52) by (4.5.17) and an easy devissage. It follows from (4.6.51) and (4.6.52) that

which contradicts (4.6.49). A a result, we have shown that

As M -⊆ M {1,2,i} for i = 3, 4, we deduce that both M min ∩ M {1,2,3} and M min ∩ M {1,2,4} are one dimensional. On the other hand, since we know that

we deduce the following direct sum decomposition

the commutative polynomial algebra T(U p ) := E[T

(j)

w ] generated by the variables T (j)

w indexed by j ∈ {1, • • • , n} and w a finite place of F over a place of Q such that ∈ D(U p ). The algebra T(U p ) acts on S(U p , E), S(U p , E) alg and S(U p , E) an via the usual double coset operators. The action of T(U p ) commutes with that of G(Q p ).

We and finally a tuple of Hodge-Tate weights k = (k 1 > k 2 > k 3 ). If ρ : Gal(F /F ) → GL 3 (E) is an absolute irreducible continuous representation which is unramified at each finite place w lying over a finite place ∈ D(U p ), we can associate to ρ a maximal ideal m ρ ⊆ T(U p ) with residual field E by the usual method described in the middle paragraph on Page 58 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. We use the notation mρ for spaces of localization and [m ρ ] for torsion subspaces where ∈ { S(U p , E), S(U p , E) alg , S(U p , E) an }.

We assume that there exists U p and ρ such that (i) ρ is absolutely irreducible and unramified at each finite place w of F over a place of Q satisfying ∈ D(U p );

(ii) S(U p , E) alg [m ρ ] = 0 (hence ρ is automorphic and ρ w0 := ρ| Gal(Fw 0 /Fw 0 ) is potentially semistable);

(iii) ρ w0 has Hodge-Tate weights k and gives the Deligne-Fontaine module D.

By identifying S(U p , E) alg with a representation of GL 3 (Q p ) via ι G,w0 , we have the following isomorphism up to normalization from (4.7.2) and [START_REF] Caraiani | Monodromy and local-global compatibility for = p[END_REF]:

(4.7.4) for all (U p , ρ) satisfying the conditions (i), (ii) and (iii), where λ = (λ 1 , λ 2 , λ 3 ) = (k 1 -2, k 2 -1, k 3 ) and d(U p , ρ) ≥ 1 is an integer depending only on U p and ρ.

Theorem 4.7.5. We consider U p = =p U and ρ : Gal(F /F ) → GL 3 (E) such that (i) ρ is absolutely irreducible and unramified at each finite place w of F lying above D(U p );

(ii) S(U p , E) alg [m ρ ] = 0;

(iii) ρ has Hodge-Tate weights k and gives the Deligne-Fontaine module D as in (4.7.3);

(iv) the filtration on D is non-critical in the sense of (ii) of Remark 6.1.4 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF];

(v) only one automorphic representation π contributes to S(U p , E) alg [m ρ ].

Then there exists a unique choice of L 1 , L 2 , L 3 ∈ E such that: We recall several useful results from [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and [START_REF] Breuil | Towards the finite slope part for GL n , prépublication[END_REF].

Proposition 4.7.7. Suppose that U p = =p U is a sufficiently small open compact subgroup of G(A ∞,p Q ), S(U p , E) an → Π Π 1 a short exact sequence of admissible locally analytic representations of GL 3 (Q p ), χ : T (Q p ) → E × a locally analytic character and η : U (t) → E its derived character, then we have T (Q p ) + -equivariant short exact sequences of finite dimensional E-spaces ( S(U p , E) an ) N (Zp) where T (Q p ) + is a submonoid of T (Q p ) defined by

Proof. This is Proposition 6.3.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] and Proposition 4.1 of [START_REF] Breuil | Towards the finite slope part for GL n , prépublication[END_REF].

Proposition 4.7.8. We fix U p and ρ as in Theorem 4.7.5. For a locally analytic character χ : Proof. This is Proposition 6.3.4 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF].

We recall the notation i GL3 B (χ ∞ w ) for a smooth principal series for each w ∈ W from Section 4.2.3. Given three locally analytic representations V i for i = 1, 2, 3 and two surjections V 1 V 2 and V 3 V 2 , we use the notation V 1 × V2 V 3 for the representation given by the fiber product of V 1 and V 3 over V 2 with natural surjections

We also use the shorten notation V alg for the maximally locally algebraic subrepresentation of a locally analytic representation V . We recall that U p is sufficiently small if there exists = p such that U has no non-trivial element with finite order.

Proposition 4.7.9. We fix U p and ρ as in Theorem 4.7.5 and assume moreover that U p is a sufficiently small open compact subgroup of G(A ∞,p Q ). We also fix a non-split short exact sequence V 1 → V 2 V 3 of finite length representations inside the category Rep OS GL3(Qp),E such that V 1 ⊗ E (ur(α) ⊗ E ε 2 ) • det embeds into S(U p , E) an [m ρ ]. We conclude that:

(i) if V 3 is irreducible and not locally algebraic, then we have an embedding

for a certain w ∈ W , then there exists a certain quotient

such that we have an embedding

Proof. This is an immediate generalization (or rather formalization) of Section 6.4 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. More precisely, part (i) (resp. (ii)) generalizes the Étape 1 (resp. the Étape 2) of Section 6.4 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF].

to finish the proof of Theorem 4.7.5. It follows from results in Section 5.3 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF] (c.f. (53) of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]) that i GL3 B (χ ∞ s1s2s1 ) has the form

and thus there is a surjection L(λ)

According to part (ii) of Proposition 4.7.9, we only need to show that any quotient V of

must have the form Σ min (λ, L 1 , L 2 , L 3 )

for certain L 3 ∈ E. We recall from Proposition 4.6.29 and our definition of Σ min (λ, L 1 , L 2 , L 3 ) afterwards that Σ min (λ, L 1 , L 2 , L 3 ) fits into a short exact sequence Σ ,+ (λ, L 1 , L 2 ) → Σ min (λ, L 1 , L 2 , L 3 ) W 0 (4.7.15)

and thus V fits (by definition of fiber product) into a short exact sequence Hence it follows from Lemma 4.6.33 that there exists L 3 ∈ E such that V ∼ = Σ min (λ, L 1 , L 2 , L 3 ).

Corollary 4.7.17. If a locally analytic representation Π of the form (4.7.11) is contained in S(U p , E) an [m ρ ] for a certain U p and ρ as in Theorem 4.7.5, then there exists L 1 , L 2 , L 3 ∈ E uniquely determined by Π such that Π → Σ min,+ (λ, L 1 , L 2 , L 3 ).

Proof. We fix U p and ρ such that the embedding and therefore we have Π ∼ = Σ Ext 1 (λ, L 1 , L 2 , L 3 ) by Theorem 6.2.1 of [START_REF] Breuil | Ext 1 localment analytique et compatibilité local-global, à paraître à[END_REF]. In particular, we deduce an embedding Π → Σ min,+ (λ, L 1 , L 2 , L 3 ) for certain invariants L 1 , L 2 , L 3 ∈ E determined by Π.