Roberto Camporesi

Jan Frahm

Roberto Miatello

Anke Pohl

Matteï Zworski D'

Bien sûr, toute thèse de doctorat commence par le remerciement de son (ses) directeur(s) de thèse. Mais ici et même avec le recul, je ne les échangerai contre aucun(s) autre(s). Angela et Tomasz on été toujours présents pour mes questions mathématiques (et rapidement). Ils ont su très brillament, et tout au long de ma thèse, me donner un peu de l'importante étendue de leurs connaissances, sans jamais me faire sentir inférieur. Aussi humainement, ils ont été là à tout moment pour m'aider à gérer ma vie extra-universitaire. Les doctorants et docteurs savent que ce n'est pas toujours facile, et parfois les avis de personnes, qui sont déjà passées par là, sont vraiment primordiaux pour prendre des décisions.

List of Figures

Introduction

The introduction in english is given just after this one.

Dans cette introduction, nous esquissons d'abord une (très) brève histoire de l'analyse harmonique, et signalons quelques jalons de l'étude de la transformée de Fourier sur les groupes de Lie. Nous donnons ensuite quelques résultats de l'analyse harmonique sur un fibré vectoriel homogène au-dessus d'un espace symétrique riemannien de type non compact. Ensuite, nous introduisons la notion de résonances du Laplacien. Après une brève présentation du contexte général motivant leur définition et leur étude, nous passons au cas du Laplacien agissant sur les sections lisses à support compact d'un fibré vectoriel homogène sur un espace symétrique riemannien X de type non-compact. Notre étude est limitée à X de rang un, mais le fibré vectoriel homogène sur X est arbitraire. La dernière section rassemble les principaux résultats de cette thèse.

1 Tour d'horizon de l'analyse harmonique En musique, les harmoniques sont les composants du son d'un instrument. Ils sont différents pour chaque instrument et sont précisément ce qui crée le timbre. Cela nous permet de distinguer différents instruments. Les sons sont des superpositions d'harmoniques, avec des fréquences multiples d'une fréquence fondamentale. Le processus de décomposition d'un son en ses harmoniques est appelé analyse harmonique.

C'est de là que vient le nom "analyse harmonique" en mathématiques. Le son est remplacé par une fonction que nous essayons de décomposer en composantes simples. Les outils les plus connus sont les extensions en série de Fourier et la transformée de Fourier. Si f est une fonction à valeurs complexes suffisamment régulière définie sur R/Z, alors

f (x) = +∞ n=-∞ c n (f )e inx .
(1)

Ici c n (f ) est le coefficient de Fourier de f défini par

c n (f ) = 1 2π 2π 0 f (t)e -int dt . ( 2
)
La formule (1) donne la décomposition en harmoniques x → c n (f )e inx .

De manière similaire, si F est une fonction "suffisamment régulière" sur R, alors

F (x) = 1 2π +∞ -∞ F (ξ)e iξx dξ , (3)
où F est la transformée de Fourier de F

F (ξ) = +∞ -∞ F (x) e -iξx dx . ( 4
)
Ici x → F (ξ)e iξx sont les "harmoniques". On arrive à "reconstruire" les fonctions f et F à partir de leurs harmoniques grâce aux formules d'inversion (1) et (3). Le théorème d'inversion (3) est valable si toutes les intégrales convergent, c'est-àdire pour F ∈ L 1 (R) et F ∈ L 1 (R). De plus, le théorème de Plancherel (1910) stipule que la transformée de Fourier, définie sur L 1 (R) ∩ L 2 (R), s'étend uniquement à un automorphisme de l'espace de Hilbert L 2 (R).

Les séries de Fourier ont été introduites par Joseph Fourier (1768 -1830) pour résoudre une équation aux dérivées partielles, connue sous le nom de problème de la "corde vibrante"

∂ 2 y ∂x 2 = 1 c 2
∂ 2 y ∂t 2 En physique, cette équation décrit le mouvement d'une corde qui vibre librement. Plus tard, en utilisant la même méthode, il a résolu l'équation de la chaleur

∂y ∂t = ∂ 2 y ∂x 2 . ( 5
)
Ces travaux peuvent être trouvés dans sa "Théorie analytique de la chaleur" [START_REF] Baptiste | Théorie Analytique de la Chaleur[END_REF].

On sait de nos jours, que nous avons ici les représentations unitaires unidimensionnelles (e n , C) du groupe commutatif R/Z sur l'espace de Hilbert C,

où n ∈ Z, R/Z × C → C (t, z) → e n (t) • z := e int • z (6)
On peut voir le n-ième coefficient de Fourier de f comme un produit de convolution :

c n (f ) = 1 2π f * e n (0) := c n (f ) = 1 2π 2π 0 f (t)e n (t, •) dt .
Ici C est appelé l'espace de la représentation (e n , C).

Une representation (π, V ) d'un groupe G sur un espace de Hilbert V est appelée unitaire, si π : G → U (V ) est un homomorphisme de groupe de G dans U (V ) des opérateurs unitaires sur V , tel que pour tout vecteur v ∈ V la fonction G g → π(g)v ∈ V est continue. Deux représentations sont dites équivalentes, s'il existe un isomorphisme de leurs espaces de Hilbert sous-jacents qui entrelace les actions de groupe correspondantes (voir par exemple [Hal15, Definition 4.3]). Une représentation est dite irréductible si aucun sous-espace de Hilbert propre non trivial de V n'est invariant sous π(G) (voir par exemple [Hal15, Definition 4.2] pour la définition exacte). On note Ĝ l'ensemble de toutes les représentations irréductibles unitaires, en identifiant les représentations isomorphes. La formule (6) montre que pour G = R/Z, Ĝ coïncide avec Z. Les coefficients de Fourier de f peuvent alors être vus aussi comme une "projection" de f sur chaque espace des représentations (6). La formule d'inversion de Fourier (1) montre que l'on peut reconstruire f en faisant la somme des projections de f sur chaque représentation irréductible.

Le cas G = R est analogue. En effet, R × C → C (t, z) → e ξ (t) • z := e iξt • z (7) est l'analogue de (6) pour R. La somme laisse place à la "somme continue" (l'intégrale) sur R et l'on voit que les deux formules (1) et (3) n'ont plus de grandes différences, sauf que le dual unitaire R est isomorphe à R lui-même.

Notons que F (ξ)e iξx = R F (t)e ξ (x -t) dt =: F * e ξ (x) .

(8)

Cela nous permet de réécrire la formule d'inversion comme suit :

F (x) = 1 2π +∞ -∞ F * e ξ (x) dξ . ( 9
)
La généralisation de l'analyse de Fourier aux groupes commutatifs localement compacts G arriva au début du XX e siècle avec les travaux de Schur, Weil, H. Cartan, 1. Tour d'horizon de l'analyse harmonique Godement et Pontryagin. Dans ce cas, Ĝ est composé de l'ensemble des homomorphismes continus G → S 1 . Ici S 1 est le cercle unité dans le plan complexe et Ĝ, munit de la multiplication des fonctions, est un groupe. On définit la transformée de Fourier par analogie pour tout f ∈ L 1 (G) par la fonction f (π) = G π(g)f (g)dg .

La transformée de Fourier peut aussi être définie sur L 2 (G). Il fut montré qu'il existe une mesure (de Haar) dν sur Ĝ, telle que la transformée de Fourier est un isomorphisme unitaire de L 2 (G) dans L 2 ( Ĝ). On a alors la formule de Plancherel

f 2 L 2 (G) = f 2 L 2 ( Ĝ) et, si f ∈ L 1 (G) et f ∈ L 1 ( Ĝ), le formule d'inversion f (g) = Ĝ π(g) f (π) dν(π) .
Cette mesure dν sur le groupe Ĝ est appelée la mesure de Plancherel.

Pour les groupes G compacts, on doit attendre le fameux théorème de Peter-Weyl (1927) qui dit que pour toute fonction f ∈ C(G) et g ∈ G, on a f (g) = π∈ Ĝ d π Tr π(g) f (π) , où f (π) = G π(g) * f (g)dg , et l'entier d π est la dimension de l'espace de la représentation π. La formule de Plancherel-Parseval

f 2 L 2 (G) = f 2 L 2 ( Ĝ)
suit. L'analyse harmonique non commutative s'est développée rapidement dans la seconde moitié du XX e siècle. La notion de transformée de Fourier a été formulée pour des groupes de groupes de type I. Cette classe comprend une très grande partie de groupes (contenant des groupes de Lie). Cependant, les résultats, principalement dus à Mautner et Segal, souffrent de leur généralité et du manque de description explicite de la mesure Plancherel sur l'objet dual Ĝ (voir [Fol16, Paragraph 7.5]). La partie principale du travail de Harish-Chandra a été de trouver cette mesure explicitement dans le cas du groupe de Lie SL(2, R) de matrices de déterminant 1 à coefficients réels (voir [START_REF] Harish-Chandra | Plancherel formula for the 2 × 2 real unimodular group[END_REF]) et plus tard pour un groupe semi-simple arbitraire. Ce fut une percée majeure dans l'analyse harmonique et les méthodes de son travail sont encore largement utilisées aujourd'hui.

Supposons à présent que G soit un groupe semi-simple réel. Harish-Chandra fit apparaître deux grandes familles de représentations dans la formule de la mesure de Plancherel. Elles sont aujourd'hui nommées les représentations de la série principale, Ĝcont , et les représentations de la série discrète, Ĝdiscr . Leurs ensembles sont notés 1. Tour d'horizon de l'analyse harmonique respectivement Ĝcont et Ĝdiscr . En ces termes, la formule abstraite de Plancherel pour L 2 (G) se présente comme suit:

L 2 (G) = Ĝcont H π ⊗ H * π dν(π) + Ĝdiscr H π ⊗ H * π . ( 10
)
On note K un sous-groupe compact maximal de G. L'espace homogène G/K est alors un espace Riemmanien symétrique de type non-compact. Il existe une décomposition du groupe G, appelée la décomposition d'Iwasawa (voir Chapitre I). Ainsi pour tout élément x dans G, il y a des éléments uniques k(x) ∈ K, a(x) ∈ A et n(x) ∈ N tels que x = k(x)a(x)n(x) . où k(x) ∈ K, a(x) ∈ A and n(x) ∈ N . Le groupe A étant commutatif, l'application "exp" définit une bijection avec son algèbre de Lie a. On peut ainsi noter a(x) = exp(H(x)). On note a * l'ensemble des formes linéaires sur l'espace vectoriel a. Un des autres nombreux travaux primordiaux de Harish-Chandra est le calcul explicite de la mesure de Plancherel dans le cas des fonctions K-bi-invariantes, soit telles que f (k 1 gk 2 ) = f (g) pour tout g ∈ G et k 1 , k 2 ∈ K. Dans ce cas, la mesure de Plancherel s'écrit à partir de la fonction c d'Harish-Chandra :

dν(λ) = c(λ) -2 dλ ,
qui a été calculée explicitement par Gindikin et Karpelevich dans [START_REF] Gindikin | Plancherel measure for symmetric Riemannian spaces of non-positive curvature[END_REF].

Helgason [START_REF] Helgason | Groups and geometric analysis, Integral geometry, invariant differential operators, and spherical functions[END_REF][START_REF] Helgason | Geometric Analysis on Symmetric Spaces[END_REF] généralisa les mêmes formules de transformées de Fourier sur les espaces G/K. Ses formules de transformées de Fourier s'écrivent comme suit. Soit M le centralisateur de A dans K. Pour des mesures bien choisies et toute fonction f sur G/K suffisamment régulières, définissons la transformée de Fourier f : a * ×K/M → C de f par

f (λ, k) = G e -(iλ+ρ)(H(g -1 k)) f (g) dg ,
où ρ ∈ a * est une constante qui dépend seulement de G et la formule d'inversion suivante est vraie :

f (x) = a * K f (λ, k)e (iλ-ρ)(H(x -1 k)) |c(λ)| -2 dλ dk .
Les analogies avec les formules précédentes sont nombreuses. En effet, si on pose la fonction sphérique φ λ (x) = K e (iλ-ρ)(H(xk)) dk , la formule d'inversion peut s'écrire comme dans les autres cas comme une convolution de f avec cette fonction sphérique,

f (x) = C G a * kM (f * φ λ )(x)|c(λ)| -2 dλ dk ,
avec C G une constante bien choisie. C'est l'analogue de (9).

Cette thèse utilise une généralisation des formules de Fourier aux sections des fibrés vectoriels au dessus de l'espace homogène G/K. Ces généralisations ont été réalisées par Camporesi dans [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF] et sont basées sur la formule de Plancherel d'Harish-Chandra sur G.

Analyse Harmonique sur les fibrés vectoriels homogènes 2 Analyse Harmonique sur les fibrés vectoriels homogènes

Un fibré vectoriel E sur une variété associe un espace vectoriel de dimension finie à chaque point de la variété. L'exemple le plus révélateur est celui du fibré tangent, qui associe à chaque point de la variété l'espace des vecteurs tangents à la variété en ce point. Par exemple, le fibré tangent de la sphère dans l'espace est la collection de tous les plans tangents à la sphère. Ces espaces vectoriels sont appelés les fibres du fibré vectoriel. Le lecteur est renvoyé à la section I.4 pour des définitions plus précises. Un fibré vectoriel E sur G/K est dit homogène si G agit sur E (par convention par la gauche) et que l'action translate les fibres (pour tout x ∈ G/K et g ∈ G, gE x = E gx ), et que cette translation soit un isomorphisme linéaire. Tout fibré vectoriel homogène est determiné par une représentation (τ, V τ ) de K. On note un tel fibré E τ . Si cette représentation n'est pas irréductible, soit V τ = V τ i , le fibré peut lui aussi être décomposé selon les composantes irrédutibles

E τ = E τ i .
On peut donc considérer que τ est irréductible (donc de dimension finie). Une section de ce fibré est une appplication qui a un point x de la variété G/K associe une vecteur de la fibre qui lui est associée. Les sections de ce fibré peuvent être vues comme les fonctions sur G à valeurs dans V τ , qui vérifient

f (xk) = τ (k -1 )f (x) pour tout x ∈ G et k ∈ K .
On note C ∞ c (G, τ ) (G ⊗ V τ ) K l'espace de ces fonctions qui sont lisses et à support compact. Contrairement aux fonctions sur G/K à valeurs complexes étudiées par Helgason, ces fonctions sont à valeurs dans un espace vectoriel avec des propriétés spécifiques sur les orbites du sous-groupe compact K. Les exemples les plus courants en physique théorique sont les formes différentielles (voir [START_REF] Pedon | Analyse harmonique des formes différentielles sur l'espace hyperbolique réel[END_REF]), les spineurs (voir [START_REF] Camporesi | Harmonic analysis for spinors on real hyperbolic spaces[END_REF]) et plus généralement n'importe quel champ tensoriel sur G/K : ces objets apparaissent naturellement dans les modèles physiques et c'est bien entendu naturel de regarder le spectre du Laplacien opérant dessus.

On considère le décomposition de la représentation τ sur M comme suit

τ = i∈I σ i , ( 11
)
où I est un ensemble fini d'indices et σ i sont des représentations irreductibles de M . Selon Camporesi, on définit la transformée de Fourier pour f ∈ C ∞ c (G, τ ) par

f (λ, k) = G e -(iλ+ρ)(H(x -1 k)) τ (k(x -1 k) -1 )f (x) dx
pour λ ∈ a * , k ∈ K. La formule d'inversion, donnée dans [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF], peut se détailler de la façon suivante. On pose P = M AN . Le sous-groupe P est dit sous-groupe parabolique minimal de G. Soit P un sous-groupe parabolique cuspidal de G tel que P ⊂ P et A ⊂ A. Soit P = M A N la décomposition de Langlands de P . On note K = M ∩ K la partie "compacte" de M et M = K A 1 N 1 la décomposition d'Iwasawa correspondante telle que A = A A 1 et N = N N 1 . Pour toute série discrète de M , il existe une représentation irréductible σ de M et µ 1 ∈ a * 1 telles que σ soit infinitésimallement équivalente à Ind M M A 1 N 1 ( σ ⊗ µ 1 ⊗ 1). Ainsi pour tout ν ∈ a * , la représentation Ind G P (σ ⊗ ν ⊗ 1) de la serie principale généralisée de P peut être vue comme une sous-représentation de Ind G P ( σ ⊗(ν +µ 1 )⊗1) et vérifie Ind G P (σ ⊗ν ⊗1)| K ⊃ τ si τ | M ⊃ σ . On a alors, pour tout f ∈ C ∞ c (G, τ ),

f (x) = 1 d τ P C P σ a * K e (iν +µ 1 -ρ)(H(x -1 k)) τ (k(x -1 k))P σ f (ν -iµ 1 , k)p σ (ν ) dν dk
où les sommes sont sur l'ensemble des sous-groupes paraboliques cuspidaux P de G décrit précédemment et sur l'ensemble des représentations σ telles que Ind G P (σ ⊗ ν ⊗ 1)| K ⊃ τ . L'opérateur P σ est la projection sur la composante isotypique de σ dans V τ . La fonction p σ est appelée la densité de Plancherel.

Si on pose ϕ σ ,ν τ la fonction sphérique définie par

ϕ σ ,ν τ = K τ (k(xk))P σ τ (k -1 )e (iν +µ 1 -ρ)(H(x -1 k)) dk , (12)
on obtient la formule d'inversion de même forme que dans les cas précédents :

f (x) = 1 d τ P C P σ a * (ϕ σ ,ν τ * f )(x)p σ (ν ) dν .
Quelques remarques :

• Si τ est la représentation triviale alors f correspond à la transformée de Helgason-Fourier sur la fonction K-invariante à droite f .

• On peut montrer que ϕ σ ,ν τ * f est une fonction lisse à support compact, si f l'est (voir Lemme I.9.3 et son corollaire).

• La formule de p σ n'est pas connue en général et c'est un problème pour la généralisation en rang supérieur. Dans cette thèse, on utilise [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF] qui la donne pour les groupes G tels que la dimension de a est 1.

Résonances

Les résonances sont des objets spectraux attachés à des opérateurs différentiels agissant sur des domaines non compacts et apparaissent comme des pôles de l'extension méromorphe de la résolvante de ces opérateurs. Ils sont liés à des objets géométriques, dynamiques et analytiques intéressants. Les questions fondamentales concernent l'existence de l'extension méromorphe de la résolvante, les propriétés de distribution et de répartition des résonances, le rang et l'interprétation des opérateurs dits résidus associés aux résonances. En conséquence, ils sont étudiés de manière intensive, dans de nombreux contextes différents, en utilisant une variété de techniques et de points de vue différents. Les références standard pour l'introduction des résonances en mathématiques sont [START_REF] Agmon | Spectral theory of Schrödinger operators on Euclidean and on non-Euclidean spaces[END_REF][START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF]. Un aperçu récent, contenant également une longue liste de références, est [START_REF] Zworski | Mathematical study of scattering resonances[END_REF]. Leur étude a dérivé de l'investigation des opérateurs de Schrödinger sur les espaces euclidiens comme R n , à une étude du Laplacien sur des espaces courbes, comme des variétés hyperboliques ou asymptotiquement hyperboliques, des espaces symétriques ou localement symétriques.

Motivations en physique quantique

Comme décrit dans l'introduction de [START_REF] Dyatlov | Mathematical theory of scattering resonances[END_REF], les résonances de diffusion ont une relation directe avec la mécanique quantique. Ils surviennent dans l'étude de la diffusion des ondes et des particules. Les physiciens ont beaucoup appris sur les forces et les interactions dans les atomes et les noyaux grâce à des expériences de diffusion. Les atomes sont bombardés de faisceaux de particules. Ensuite, ces particules sont détectées par des dispositifs qui donnent l'intensité en fonction de l'angle de diffusion ou de l'énergie. Le Laplacien positif -∆ sur L 2 (R d ) représente en mécanique quantique l'opérateur d'énergie cinétique d'une particule quantique libre. Les nombreuses interactions, par exemple qui se produisent dans une expérience de diffusion, sont représentées par une fonction à valeur réelle, appelée potentiel V ∈ L ∞ c (R d ). Nous supposons sans perte de généralité que V a un support compact, car à l'infini les interactions avec le centre de diffusion sont nulles. L'opérateur d'énergie dans ce cas n'est rien d'autre qu'un opérateur de Schrödinger, également appelé hamiltonien perturbé (perturbé par V )

H V = -∆ + V .
(13) D'après le théorème de Kato-Rellich, H V est auto-adjoint. Son spectre est donc réel. Sa résolvante

R(λ 2 ) = (H V -λ 2 ) -1
est bien définie sur le demi-plan supérieur (λ) > 0 comme un opérateur borné sur L 2 (R). Lorsqu'il est restreint à C ∞ c (R), R(λ 2 ) s'étend à une fonction méromorphe de λ sur le plan complexe. Les singularités de la résolvante étendue sont appelées les résonances de cet opérateur de Schrödinger. En général, il est très difficile de les calculer. C'est possible dans des cas très particuliers, et pour un potentiel très particulier. Un cas bien connu est celui de l'opérateur de Laplace sur R n (voir section II.1.1). Le but de cette thèse est de faire les mêmes calculs, en remplaçant R n par un fibré vectoriel homogène (sur un espace symétrique de type non compact et de rang réel un).

Faisons un calcul plus concret lié à l'équation d'onde. En fait, la solution de cette équation a un taux d'oscillation et de décroissance donné par les résonances de diffusion. En dimension un, l'équation d'onde est donnée par

(∂ 2 t + H V ) u(t, x) = (∂ 2 t -∂ 2 x + V (x)) u(t, x) = 0 .

Résonances

Supposons les conditions initiales u(0, x) = u 0 (x) et ∂ t w(0, x) = w 1 (x), où w j ∈ C ∞ c (R). Alors pour tout A > 0, on a l'expansion de résonance (séries de Fourier dans le cas où V = 0)

w(t, x) = λ j >-A e -ilλx f λ,l (x) + E A (t, x) ,
où les λ j varient sur l'ensemble des pôles de la résolvante étendue R(λ), la somme est finie, (H V -λ 2 ) l+1 f λ,l = 0 et le terme d'erreur E A (t, x) est borné par e -At dans un sens bien défini, voir [DZ19, Théorème 2.9].

Les pôles λ j , avec (λ j ) < 0 de la résolvante étendue sont appelés résonances de diffusion ou simplement résonances de l'opérateur H V . Les simulations numériques des solutions montrent que ces fonctions ont des pics décroissant à l'infini. On remarque que la partie réelle des résonances est la vitesse des oscillations et la partie imaginaire est l'opposé de la vitesse de la décroissance. La théorie qui en résulte attire de nombreux chercheurs.

Bien sûr, il s'agit d'un bref aperçu de l'utilité des résonances dans un cadre physique. Pour des applications dans différentes situations, voir [DZ19, Chapitre 1].

Résonances de l'opérateur de Laplace sur les espaces symétriques riemanniens non compacts

Les espaces symétriques riemanniens de type non compact ont des structures géométriques importantes pour étudier les résonances du Laplacien. En plus d'être des objets intrinsèquement intéressants, ils jouent le rôle d'espaces modèles pour comprendre des phénomènes sur des géométries plus compliquées ou moins régulières. Dans un cadre typique, on travaille sur une variété riemannienne complète X de géométrie finie, pour laquelle le Laplacien positif ∆ est un opérateur essentiellement auto-adjoint sur l'espace de Hilbert L 2 (X) des fonctions de carré intégrable sur X. On suppose que ∆ a un spectre continu [ρ X , +∞[, avec ρ X ≥ 0. Le spectre de ∆ peut avoir des parties discrètes, mais ces parties ne jouent aucun rôle significatif dans la suite; nous les négligeons donc. Aussi, pour simplifier les explications, on translate le Laplacien de sorte que son spectre commence à 0, on change les variables z → z 2 pour la résolvante de sorte que, comme ci-dessus, la résolvante soit analytique en dehors de l'axe réel. La résolvante R(z) = (∆ -ρ X -z 2 ) -1 du Laplacien translaté ∆ -ρ X est alors une fonction holomorphe de z sur le demi-plan complexe supérieur (et sur le demi-plan complexe inférieur).

Pour chacun de ces complexes z, R(z) est un opérateur linéaire borné de L 2 (X) dans lui-même. En tant que tel, il ne peut pas être étendu sur l'axe réel. Limitons la résolvante au sous-espace dense C ∞ c (X) des fonctions lisses à support compact sur X. Sur cet espace, l'application z → R(z) peut admettre une extension méromorphe à travers l'axe réel vers un domaine plus grand dans C (vers une surface de Riemann au dessus de C). Les pôles de cet extension méromorphe, s'ils existent, sont les résonances, également appelées résonances quantiques ou pôles de diffusion (comme expliqué dans le paragraphe précédent), de ∆.

Introduisons quelques notations. Un espace symétrique riemannien de type non compact est un espace homogène de la forme X = G/K où G est un groupe de Lie semi-simple réel non compact connexe de centre fini et K est un sous-groupe compact maximal de G. Les exemples de base sont les espaces hyperboliques réels ndimensionnels H n . Dans ce cas, le groupe de Lie G est le groupe de Lorentz généralisé SO e (n, 1) et K = SO(n). Un espace symétrique riemannien de type non compact X a des sous-espaces plats maximaux, tous de même dimension, appelée rang (réel) de X. Par exemple, le rang de H n est 1. Puisque X est un espace symétrique du groupe de Lie G, tous les opérateurs naturels agissant sur X, comme le Laplacien et sa résolvante, sont G-invariants. Ils peuvent donc être étudiés en utilisant la théorie des représentations de G. L'étude analytique de la résolvante du Laplacien agissant sur les fonctions sur H n , en particulier son extension méromorphe à travers son spectre, est classique et bien comprise. Elle joue un rôle central dans l'étude de la résolvante sur des variétés riemanniennes complètes plus générales pour lesquelles les espaces hyperboliques sont des modèles. Toujours dans le cas des fonctions sur un espace symétrique riemannien plus général X = G/K, l'étude des résonances peut en principe se faire à l'aide d'une analyse harmonique adaptée, dite analyse d'Helgason-Fourier, qui fournit une diagonalisation du Laplacien et donc une formule explicite pour sa résolvante en tant qu'opérateur intégral singulier sur le spectre. Cette formule a permis à Hilgert et Pasquale [START_REF] Hilgert | Resonances and residue operators for symmetric spaces of rank one[END_REF] de déterminer et d'étudier les résonances pour un X arbitraire de rang un. Le cas général de rang supérieur est toujours ouvert. Le cas de rang un a été précédemment résolu par Miatello et Will [START_REF] Miatello | The residues of the resolvent on Damek-Ricci spaces[END_REF] avec une méthode différente, dans le contexte des espaces de Damek-Ricci. D'autres travaux pertinents dans ce contexte sont [START_REF] Mazzeo | Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type[END_REF] et [START_REF] Strohmaier | Analytic continuation of resolvent kernels on noncompact symmetric spaces[END_REF]. Les réponses aux problèmes de base concernant l'existence et la localisation des résonances du Laplacien, ainsi que l'interprétation théorique de la représentation des opérateurs dits résiduels aux résonances, ne sont connus que pour (la plupart) des espaces symétriques riemanniens de rang 2. Ces résultats sont apparus dans des articles communs de Hilgert, Pasquale et Przebinda (voir [START_REF] Hilgert | Resonances for the laplacian on products of two rank one riemannian symmetric spaces[END_REF][START_REF] Hilgert | Resonances for the Laplacian on Riemannian symmetric spaces: the case of SL(3, R)/SO(3)[END_REF][START_REF] Hilgert | Resonances for the Laplacian: the cases BC 2 and C 2 (except SO 0 (p, 2) with p > 2 odd)[END_REF]).

Tous les articles mentionnés ci-dessus considèrent le Laplacien agissant sur des fonctions scalaires sur X. Une question plus générale est de considérer le Laplacien agissant sur des sections d'un fibré vectoriel homogène sur X. Un tel fibré est déterminé par une représentation de dimension finie τ de K. Notons ce fibré E τ . Les sections de E τ peuvent être vues comme des fonctions vectorielles sur G, à valeurs dans l'espace de la représentation τ , telles que

f (xk) = τ (k -1 )f (x) pour tout x ∈ G et k ∈ K .
L'espace de ces fonctions qui sont lisses et à support compact est noté C ∞ c (G, τ ). Cela signifie que nous remplaçons les fonctions à valeurs complexes sur X par des fonctions à valeurs vectorielles qui ont des propriétés de transformations spécifiques sur les orbites du sous-groupe compact K. Ceci est une généralisation de l'étude précédente, dans le sens où les sections du fibré vectoriel homogène E triv K , pour la représentation triviale de K, sont exactement les fonctions à valeurs complexes sur X.

Des exemples de sections de fibrés vectoriels homogènes sur X sont les formes différentielles, les champs vectoriels, et plus généralement les champs tensoriels sur X: tous ces objets apparaissent naturellement dans les modèles physiques, et il est donc naturel de rechercher des résonances du Laplacien dans ces contextes.

La résolvante du Laplacien des formes sur un espace symétrique riemannien de rang un de type non-compact a été étudiée par plusieurs auteurs; voir [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF][START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over compact Riemannian symmetric spaces-the local theory[END_REF][START_REF] Camporesi | A generalization of the Cartan-Helgason theorem for Riemannian symmetric spaces of rank one[END_REF][START_REF] Camporesi | The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF][START_REF] Pedon | Analyse harmonique des formes différentielles sur l'espace hyperbolique réel[END_REF][START_REF] Carron | On the differential form spectrum of hyperbolic manifolds[END_REF][START_REF] Bunke | The spectrum of Kleinian manifolds[END_REF]. En particulier, [START_REF] Carron | On the differential form spectrum of hyperbolic manifolds[END_REF] donne (pour les formes différentielles) la liste des résonances et la surface de Riemann sur laquelle la résolvante admet l'extension méromorphe. Néanmoins, à notre connaissance, il n'existe qu'un seul article étudiant les résonances et les opérateurs résiduels du Laplacien agissant sur les sections d'un fibré vectoriel homogène sur X, à savoir [START_REF] Will | The meromorphic continuation of the resolvent of the Laplacian on line bundles over CH(n)[END_REF], où X est l'espace hyperbolique complexe et les fibres sont de dimension 1.

Résultats de la thèse

Le but de cette thèse est d'étudier les résonances du Laplacien agissant sur les sections d'un fibré vectoriel homogène sur un espace symétrique riemannien de type non compact X = G/K. L'espace symétrique est supposé avoir un rang réel égal à un mais la représentation de K est arbitraire. Puisque toute représentation de dimension finie de K se décompose en irréductibles, nous restreignons notre attention aux représentations irréductibles. Comme dans le cas des fonctions, les problèmes que nous soulevons dans cette thèse, sont de déterminer l'existence et la localisation des résonances et d'étudier les opérateurs de résidus qui leur sont associés.

Cette thèse est organisée comme suit. Dans le chapitre I, nous introduisons les notations et rappelons quelques faits de base sur la structure des espaces symétriques riemanniens de type non compact et de rang réel un. Il y a quatre cas, répertoriés dans le tableau suivant:

G K X = G/K Spin(n, 1) Spin(n) espace hyperbolique réel SU(n, 1) S(U(n) × U(1))
espace hyperbolique complexe Sp(n, 1) Sp(n) espace hyperbolique quaternionique F 4 Spin(9) espace hyperbolique octonionique On pose a comme un sous-espace plat maximal dans p = T eK (X) l'espace tangent de X au point de base eK, et M le centralisateur de a dans K. Dans la section I.9, nous rappelons quelques faits sur la généralisation de la transformée d'Helgason-Fourier aux fibrés vectoriels homogènes. Ils sont principalement dus à Camporesi (voir [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF]). En particulier, le théorème de Plancherel pour les sections L 2 des fibrés vectoriels homogènes y est donné. Désignons la décomposition de τ sur M comme suit :

τ = σ∈ M (τ ) d σ σ (14)
où M (τ ) est l'ensemble des représentations unitaires irréductibles de M qui apparaissent dans la restriction de τ à M , d σ est le degré de σ. Nous avons besoin de certaines propriétés des fonctions sphériques généralisées ϕ σ,λ τ associées aux représentations irréductibles σ ∈ M (τ ), qui sont détaillées dans la section I.9. La formule explicite de la densité de Plancherel p σ correspondant à ces σ ∈ M est donnée dans la proposition II.1.4 (voir aussi l'annexe A). Le corollaire I.9.1 prouve la convergence de l'opérateur intégral singulier fournissant une formule explicite pour la résolvante R du Laplacien en utilisant la formule d'inversion de la transformée de Helgason-Fourier à valeurs vectorielles. Dans la section II.1, nous calculons les résonances, ce qui est le premier objectif principal de cette thèse. La fonction holomorphe z → R(z) est étendue méromorphiquement du demi-plan supérieur complexe à l'espace entier, en utilisant le théorème des résidus. La résolvante étendue est une fonction méromorphe avec des pôles simples sur l'axe imaginaire: ces pôles sont les résonances. Cela conduit à notre premier théorème :

Theorem 1 Soit G un groupe de Lie semi-simple non compact connexe de centre fini et de décomposition d'Iwasawa G = KAN , où K est un sous-groupe compact maximal fixe de G.

Supposons que dim A = 1. Soit M le centralisateur de A en K. Soit (τ, H τ ) une représentation unitaire irréductible de K, et soit E τ le fibré vectoriel homogène sur G associé à τ . Pour chaque σ ∈ M (τ ), soit N σ l'ensemble de k ∈ Z tel que λ k := -i(B max + k)
est un pôle de la densité de Plancherel (voir (II.15) pour la formule) et B max + k ≥ 0. Ici B max est une constante non négative qui ne dépend que de G et σ. On se réfère à (II.17), (II.18) et (II.20) pour la définition précise.

Dans ce cadre, la continuation méromorphe de la résolvante R de l'opérateur de Laplace agissant sur les sections lisses et à support compact de E τ peut être écrit comme la somme

R = σ∈ M (τ ) d σ R σ , ( 15
) où R σ est donné pour tout f ∈ C ∞ c (G, τ ) et pour tout N ∈ N par la formule suivante: (R σ (ζ σ )f ) (x) = 1 |α| R-i(N+1/4) 1 ζ σ -λ|α| ϕ σ,λα τ * f (x) p σ (λα) λ dλ + 2iπ |α| k∈Nσ λ k >-i(N +1/4) 1 ζ σ -λ k |α| ϕ σ,λ k α τ * f (x) Res λ=λ k p σ (λα) λ (16)
Dans (16), α est la racine restreinte la plus longue, ϕ λ,σ τ est une fonction sphérique de type σ et

ζ σ := -z -ρ, ρ + µ σ + ρ M , µ σ + ρ M ) (17) avec z ∈ C tel que (ζ σ ) > -(N + 1/4). Ici, √
• désigne la branche à valeur unique de la fonction racine carrée déterminée sur C \ [0, +∞[ par la condition √ -1 = -i.

Les résonances de l'opérateur de Laplace agissant sur les sections de E τ apparaissent dans des familles paramétrées par les éléments de M (τ ). Soit

S σ = (z, ζ) ∈ C 2 | ζ 2 := -z -ρ, ρ + µ σ + ρ M , µ σ + ρ M ) . Alors la resolvante R σ s'étend méromorphiquement de S + σ = {(z, ζ) ∈ S | (ζ) > 0} à S σ .
Les pôles (simples) de cette extension sont les couples

(z σ,k , ζ σ,k ) = (B max + k) 2 |α| 2 -ρ 2 α |α| 2 + µ σ + ρ M , µ σ + ρ M , -i(B max + k) 2 |α| 2 (18)
où µ σ est le plus haut poids de la représentation σ, les nombres k sont dans N σ et ρ M est la demi-somme des racines pour M .

Nous renvoyons à la section I pour les définitions des différents objets apparaissant dans ce théorème et à la section II.1 pour sa démonstration.

Le deuxième problème que nous abordons dans cette thèse est l'interprétation théorique des représentations issues des résonances. Plus précisément, considérons la partie résiduelle de la suite méromorphe de la résolvante dans (16). Pour chaque pôle λ k de la densité de Plancherel pour σ ∈ M (τ ), on peut introduire un opérateur, appelé opérateur résiduel en λ k α, défini comme suit:

R σ k : C ∞ c (G, τ ) -→ C ∞ (G, τ ) f -→ ϕ σ,λ k α τ * f (19)
Puisque le produit de convolution est à gauche, il ne permute pas avec la translation à gauche de f (voir (I.22)). Comme G agit sur l'image de R σ k par translation à gauche, nous obtenons une représentation de G, appelée représentation résiduelle en λ k α.

Dans la section II.2 nous limitons notre attention aux représentations τ qui contiennent la représentation triviale de M . Dans ce cas, la structure de la série principale est bien connue (voir [START_REF] Howe | Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations[END_REF][START_REF] Johnson | Composition series and intertwining operators for the spherical principal series[END_REF][START_REF] Johnson | Composition series and intertwining operators for the spherical principal series[END_REF]). La complexité du cas général (voir [START_REF] David | Representations of rank one Lie groups[END_REF]) est particulièrement intéressante et pourrait conduire à des résultats beaucoup moins généralisables. Nous l'évitons donc dans cette thèse. On considère les résonances correspondant à σ = triv. Pour simplifier la notation, nous écrivons R k au lieu de R triv k . Soit E k la représentation résiduelle en λ k α. Nous montrons que les E k sont irréductibles et équivalentes à un sous-quotient d'une représentation de la série principale sphérique de G. Nous déterminons lesquelles d'entre elles sont unitaires et lesquelles sont de dimension finie. De plus, nous identifions leurs paramètres de Langlands et calculons leurs fronts d'onde. Les paramètres de Langlands sont de la forme (M A, δ, ν) et désignent la représentation induite Ind G M AN (δ ⊗ e iν ⊗ triv). Un K-type minimal de la représentation induite de plus haut poids µ min identifie un sous-quotient irréductible unique de cette représentation induite (voir [START_REF] Vogan | Classification of the irreducible representations of semisimple Lie groups[END_REF]). L'ensemble du front d'onde d'une représentation a été introduit par Howe (voir [START_REF] Howe | Wave front sets of representations of Lie groups[END_REF]). Lorsque G est semi-simple, c'est un ensemble fermé constitué d'orbites nilpotentes. Pour E k , il s'agit de la fermeture d'une seule orbite nilpotente. Dans le théorème suivant, α est la plus longue racine restreinte comme dans le théorème 1 et pour chaque racine restreinte β, l'espace correspondant dans g est noté g β . Le K-type minimal de E k est donné dans la démonstration du théorème dans chaque cas: on les retrouve dans les tables II.1, II.2, II.3 et II.4 respectivement pour les espaces hyperboliques réel, complexe, quaternionique et octonionique.

Theorem 2

Supposons que la représentation τ contienne la représentation triviale de M . Les représentations de résidus E k sont alors irréductibles.

1. Si G = Spin(2n, 1), alors τ a un plus haut poids de la forme (N, 0, . . . , 0), où N est un entier non négatif.

• Si N ≥ k+1, alors E k a les paramètres de Langlands M A, H k+1 (R 2n-1 ), n -3 2 α avec (k+1, 0, . . . , 0) comme plus haut poids du K-type minimal. Ici H k+1 (R 2n-1 ) sont les polynômes harmoniques de degré k + 1 sur R 2n-1 . Cette représentation est unitaire. Son ensemble de front d'onde est l'orbite nilpotente générée par g α .

• Si N < k + 1, alors E k a le paramètre de Langlands M A, triv, (ρ α + k) avec la représentation triviale comme K-type minimal. Il est de dimension finie. De plus, il n'est pas unitaire si k = 0.

2. Si G = SU(n, 1), alors τ a un plus haut poids de la forme (M 1 , 0, . . . , 0, -M 2 , -L),

où M 1 et M 2 sont des entiers positifs tels que M 1 ≥ M 2 ≥ 0, L ∈ Z et M 1 +M 2 +L est pair. • Si M 1 + M 2 ≥ 2k + 2 et |L| ≤ -2k -2 + M 1 + M 2 , alors E k est unitaire.
si n > 2, alors E k a un K-type minimal de plus haut poids (k + 1), 0, . . . , 0, -(k+1), 0). Son paramètre de Langlands est M A, δ, n 2 -1 α où le plus haut poids de δ est (k + 1), 0, . . . , 0, -(k + 1), 0 . Son front d'onde est l'orbite nilpotente générée par g α/2 . -si n = 2, cette représentation est la série discrète avec le paramètre Blattner ((k + 1), -(k + 1), 0, . . . , 0) . Son front d'onde est l'orbite nilpotente générée par g α/2 .

• Si L ≥ | -2k -1 + M 1 + M 2 | + 1,
• Si L ≤ | -2k -1 + M 1 + M 2 | -1,
• Si M 1 + M 2 ∈ [0, 2k + 2[ et L < |2k + 2 -M 1 + M 2 |,

Introduction (English version)

In this introduction, we first sketch a (very) brief history of harmonic analysis, and point out some milestones on the study of the Fourier transform on Lie groups. We then outline some results in the harmonic analysis on a homogeneous vector bundle over a Riemannian symmetric space of non-compact type. Next, we introduce the notion of resonances of the Laplacian. After a short presentation of the general context motivating their definition and study we pass to case of the Laplacian acting on the compactly supported smooth sections of a homogeneous vector bundle over a Riemannian symmetric space X of non-compact type. Our study is restricted to X of rank one , but the homogeneous vector bundle over X is arbitrary. The last section collects the main results of this thesis.

A brief overview of harmonic analysis

In music, harmonics are the components of the sound of an instrument. They are different for each instrument and are precisely what creates the timbre. This allows us to distinguish different instruments. Sounds are superpositions of harmonics, with frequencies that are multiples of a fundamental frequency. The process of breaking down a sound into its harmonics is referred to as harmonic analysis. This is where the name "harmonic analysis" in mathematical originates. The sound is replaced by a function that we try to break down into simple components. The best known tools are expansions in Fourier series and Fourier transform. If f is a sufficiently regular complex-valued function defined on R/Z, then

f (x) = +∞ n=-∞ c n (f )e inx .
(1)

Here c n (f ) is the Fourier coefficient of f , defined by

c n (f ) = 1 2π 2π 0 f (t)e -int dt . (2)
Formula (1) gives the decomposition into its harmonics

x → c n (f )e inx .
Similarly, if F is a sufficiently regular, complex valued function on R, then

F (x) = 1 2π +∞ -∞ F (ξ)e iξx dξ , ( 3
)
where F is the Fourier transform of F

F (ξ) = +∞ -∞ F (x) e -iξx dx . ( 4
)
Here the harmonics are x → F (ξ)e iξx . We manage to "reconstruct" the functions f and F from their harmonics thanks to the inversion formulas (1) and (3). The inversion theorem (3) holds if all integrals converge, i.e. for F ∈ L 1 (R) and F ∈ L 1 (R). Moreover, Plancherel's theorem (1910) states that the Fourier transform, defined on L 1 (R) ∩ L 2 (R), uniquely extends to a Hilbert space isomorphism of L 2 (R) in itself. The Fourier series were introduced by Joseph Fourier (1768 -1830) to solve a partial differential equation, known as the "vibrating string" problem

∂ 2 y ∂t 2 = c 2 ∂ 2 y ∂x 2 .
Physically, this equation describes the motion of a freely vibrating string. Later, using the same method, he solved the heat equation

∂y ∂t = ∂ 2 y ∂x 2 . ( 5
)
1. A brief overview of harmonic analysis These works can be found in his "Analytical Theory of Heat" [START_REF] Baptiste | Théorie Analytique de la Chaleur[END_REF].

In contemporary terms, what we have here are the one dimensional unitary representations (e n , C) of the commutative group R/Z on the Hilbert space C, indexed by

n ∈ Z, R/Z × C → C (t, z) → e n (t) • z := e int • z . ( 6
)
Here C is called the space of the representation (e n , C).

A representation (π, V ) of a group G on a Hilbert space V is called unitary, if

π : G → U (V )
is a group homomorphism from G into the group U (V ) of the unitary operators on V , such that for every vector v ∈ V the function G g → π(g)v ∈ V is continuous. Two representations are said to be equivalent, if there exists an isomorphism of their underlying Hilbert spaces which intertwines the corresponding group actions (see for example [Hal15, Definition 4.3]). A representation is said to be irreducible if no nontrivial proper Hilbert subspace of V is invariant under π(G) (see for example [START_REF] Hall | Lie groups, Lie algebras, and representations, An elementary introduction[END_REF]Definition 4.2] for the exact definition). We denote by Ĝ the set of all equivalence classes of the irreducible unitary representations of G. The formula (6) shows that for G = R/Z, Ĝ coincides with the integers. Fourier inversion (1) shows that we can reconstruct f by adding these projections of f on each irreducible representation.

The case

G = R is analogous. Indeed, R × C → C (t, z) → e ξ (t) • z := e iξt • z (7)
is the analogue of (6) for R. The sum gives way to the "continuous sum" (the integral) over R and we see that the two formulas (1) and (3) have no longer significant differences, except that the unitary dual R is identified with R. Notice that

F (ξ)e iξx = R F (t)e ξ (x -t) dt =: F * e ξ (x) . ( 8
)
This allows us to rewrite the inversion formula as

F (x) = 1 2π +∞ -∞ F * e ξ (x) dξ . ( 9
)
The generalization of Fourier analysis to a commutative locally compact group G arrived at the beginning of the XXth century with the works of Schur, Weil, H. Cartan, Godement and Pontryagin. In this case Ĝ consists of continuous homorphisms G → S 1 . Here S 1 is the unit circle in the complex plane and Ĝ, with multiplication of functions, is a group. By analogy, one defines the Fourier transform of f ∈ L 1 (G) as the function of Ĝ given by f

(π) = G π(g)f (g)dg .
The Fourier transform can also be defined on L 2 (G). It was shown that there is a Haar measure dν on Ĝ, such that the Fourier transform is a unitary isomorphism of L 2 (G) onto L 2 ( Ĝ). We have the Plancherel formula

f 2 L 2 (G) = f 2 L 2 ( Ĝ)
and, if f ∈ L 1 (G) and f ∈ L 1 ( Ĝ), the inversion formula

f (g) = Ĝ π(g) f (π) dν(π) .
One refers to this dν as to the the Plancherel measure.

For compact G groups (not necessarily commutative) , we had to wait for the famous Peter-Weyl theorem (1927), which says that for any function f ∈ C(G) and g ∈ G,

f (g) = π∈ Ĝ d π Tr π(g) f (π) , where f (π) = G π(g) * f (g)dg ,
and the integer d π is the dimension of the space of the representation π. The Plancherel-Parseval formula

f 2 L 2 (G) = f 2 L 2 ( Ĝ)
follows. The non-commutative harmonic analysis developped rapidly in the second half of the twentieth century. The notion of the Fourier transform was formulated for groups of type I. This class includes a very large part of groups (containing Lie groups). However, the results, mainly due to Mautner and Segal, suffer from its generality and lack of explicit description of the Plancherel measure on the dual object Ĝ (see [START_REF] Folland | A course in abstract harmonic analysis[END_REF]Paragraph 7.5]). The main part of Harish-Chandra's work was to find this measure explicitly in the case of the Lie group SL(2, R) of matrices of determinant 1 with real coefficients (see [START_REF] Harish-Chandra | Plancherel formula for the 2 × 2 real unimodular group[END_REF]) and later for an arbitrary real semisimple group. This was a major breakthrough in harmonic analysis and the methods of his work are still widely used today.

From now on we assume that G is a real semi-simple group with a finite center. Harish-Chandra revealed two large families of representations which appear in the Plancherel measure. They are called representations of the principal series, Ĝcont , and the representations of the discrete series, Ĝdiscr . In these terms the abstract Plancherel formula for L 2 (G) looks as follows,

L 2 (G) = Ĝcont H π ⊗ H * π dν(π) + Ĝdiscr H π ⊗ H * π . ( 10
)
Denote by K a maximal compact subgroup of G. The homogeneous space G/K is called a Riemmanian symmetric space of non-compact type. There exists a decomposition of the group G, G = KAN , called the Iwasawa decomposition (see Chapter I). Thus for any element x in G, we have unique elements k(x) ∈ K, a(x) ∈ A and n(x) ∈ N such that x = k(x)a(x)n(x) .

The map exp : a → A, from the Lie algebra to the group, is a bijection. Hence there is a unique H(x) ∈ a such that a(x) = exp(H(x)). We denote by a * the space of linear forms on vector space a. One of Harish-Chandra's early works is the Plancherel theorem for the space of square integrable K-bi-invariant functions on G (i.e. f (k 1 gk 2 ) = f (g) for all g ∈ G and k 1 , k 2 ∈ K). In this case, the Plancherel measure is written in term of Harish-Chandra's c function,

dν(λ) = |c(λ)| -2 dλ ,
and was explicitly computed by Gindikin and Karpelevich in [START_REF] Gindikin | Plancherel measure for symmetric Riemannian spaces of non-positive curvature[END_REF].

Helgason [START_REF] Helgason | Groups and geometric analysis, Integral geometry, invariant differential operators, and spherical functions[END_REF][START_REF] Helgason | Geometric Analysis on Symmetric Spaces[END_REF] generalized the Plancherel theorem from the K-bi-invariant functions to functions defined on the symmetric space G/K. His Fourier transform formulas looks as follows. Let M be the centralizer of A in K. For suitably normalized measures and sufficiently regular functions f on G/K, define the Helgason-Fourier

transform f : a * × K/M → C by f (λ, k) = G e -(iλ+ρ)(H(g -1 k)) f (g) dg ,
where ρ ∈ a * depends only on G, and the following inversion formula is true:

f (x) = a * k f (λ, k)e (iλ-ρ)(H(x -1 k)) |c(λ)| -2 dλ d(k) .
There are similarities with the preceding Euclidean formulas. Indeed, if we define the spherical function

φ λ (x) = K e (iλ-ρ)(H(xk)) dk ,
then the inversion formula can also be written in terms of a convolution of f with this spherical function,

f (x) = a * K (f * φ λ )(x)|c(λ)| -2 dλ dk .

This is the analog of (9).

This thesis uses a generalization of Helgason's Fourier formulas to sections of vector bundles over the homogeneous space G/K. These generalizations are due to Camporesi (see [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF]) and are based on Harish-Chandra's Plancherel formula on G.

Harmonic analysis on homogeneous vector bundles

A vector bundle E over a manifold associates a vector space with each point of the manifold. The most telling example is that of the tangent bundle, which associates with each point of the manifold the space of vectors tangent to the manifold at this point. For example, the tangent bundle of the sphere in R 3 is the collection of all planes tangent to the sphere. These vector spaces are called the fibers of the vector bundle. The reader is referred to the I.4 for more precise definitions.

A vector bundle E over G/K is called homogeneous if G acts on E (by convention on the left) and the action translates the fibers (for all x ∈ G/K and g ∈ G, gE x = E gx ), and each translation is a linear isomorphism. Any homogeneous vector bundle is determined by a representation (τ, V τ ) of K. We denote a such bundle by E τ . If V τ = V τ i is a sum of irreducible components, then the bundle can also be decomposed,

E τ = E τ i .
We focus on the case when τ is irreducible (hence of finite dimension). A section of the vector bundle is a function which to a point x ∈ G/K associates a vector in the fiber over it. The sections can be seen as functions on G with values in V τ satisfying the following equation,

f (xk) = τ (k -1 )f (x) for all x ∈ G and k ∈ K . We denote by C ∞ c (G, τ ) (C ∞ c (G) ⊗ V τ )
K the space of smooth and compactly supported functions as above. Compared to the complex valued functions on G/K studied by Helgason, these functions are valued in a vector space and have the above specified properties under the action of the compact subgroup K by right translations. The most common examples in theoretical physics are differential forms (see [START_REF] Pedon | Analyse harmonique des formes différentielles sur l'espace hyperbolique réel[END_REF]), spinors (see [START_REF] Camporesi | Harmonic analysis for spinors on real hyperbolic spaces[END_REF]) and more generally any tensor field on G/K: these objects appear in physical models and it is natural to study the spectrum of the Laplacian operating on them.

We consider the restriction of the representation τ to M ,

τ | M = i∈I σ i , ( 11
)
where I is a finite index set and the σ i are irreducible representations of M . Following Camporesi, we define the Fourier transform of

f ∈ C ∞ c (G, τ ) by f (λ, k) = G e -(iλ+ρ)(H(x -1 k)) τ (k(x -1 k) -1 )f (x) dx ,
where λ ∈ a * and k ∈ K. The inversion formula, given in [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF], can be explained as follows. We set P = M AN . The subgroup P is called a minimal parabolic subgroup of G. Let P be a cuspidal parabolic subgroup of G such that P ⊂ P and A ⊂ A.

Let P = M A N be the Langlands decomposition of P . We denote by

K = M ∩ K the "compact" part M and M = K A 1 N 1 the corresponding Iwasawa decomposition such that A = A A 1 and N = N N 1 . For any discrete series of M , there is an irreducible representation σ of M and µ 1 ∈ a * 1 such that σ is infinitesimally equivalent to Ind M M A 1 N 1 ( σ ⊗ µ 1 ⊗ 1)
. Thus for all ν ∈ a * , the representation Ind G P (σ ⊗ ν ⊗ 1) of the generalized principal series of P can be seen as a subrepresentation of Ind G P ( σ

⊗ ν + µ 1 ⊗ 1) and verify Ind G P (σ ⊗ ν ⊗ 1)| K ⊃ τ if τ | M ⊃ σ . So we have, for all f ∈ C ∞ c (G, τ ), f (x) = 1 d τ P C P σ a * K e (iν +µ 1 -ρ)(H(x -1 k)) τ (k(x -1 k))P σ f (ν -iµ 1 , k)p σ (ν ) dν dk
where the sums are over the set of cuspidal parabolic subgroups P of G described previously and over the set of representations σ such that Ind G P (σ ⊗ ν ⊗ 1)| K ⊃ τ . The operator P σ is the projection on the isotypic component of σ in V τ . The function p σ is called the Plancherel density.

If we denote by ϕ σ ,ν τ the corresponding spherical function

ϕ σ ,ν τ = K τ (k(xk))P σ τ (k -1 )e (iν +µ 1 -ρ)(H(x -1 k)) dk , ( 12
)
we obtain the same form for the inversion formula as in the previous cases:

f (x) = 1 d τ P C P σ a * (ϕ σ ,ν τ * f )(x)p σ (ν ) dν .
Some remarks:

• If τ is the trivial representation then f agrees with the Helgason-Fourier transform of the right-K-invariant function f .

• If f is a smooth function with compact support then ϕ σ ,ν τ * f is a Paley-Wiener function (see Lemma I.9.3 and its Corollary).

• The formula of p σ is not known in general and this causes problems. In this thesis, we use [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF] which gives the formula for groups G of real rank one, that is the dimension of a is 1.

Resonances

Resonances are spectral objects attached to differential operators acting on non-compact domains and appear as poles of the meromorphic continuation of the resolvent of these operators. They are linked to interesting geometric, dynamical, and analytic objects.

The basic questions concern the existence of the meromorphic extension of the resolvent, the distribution and counting properties of the resonances, the rank and interpretation of the so-called residue operators associated with the resonances. As a consequence, they are intensively studied, in many different settings, using a variety of techniques and different viewpoints. Standard references for the introduction of resonances in mathematics are [START_REF] Agmon | Spectral theory of Schrödinger operators on Euclidean and on non-Euclidean spaces[END_REF][START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF]. A recent overview, also containing an extensive list of references, is [START_REF] Zworski | Mathematical study of scattering resonances[END_REF]. Their study evolved from an investigation of the Schrödinger operators on the Euclidean spaces like R n , to a study of the Laplacian on curved spaces, like hyperbolic or asymptotically hyperbolic manifolds, symmetric or locally symmetric spaces.

Motivations from quantum physics

As described in the introduction of [START_REF] Dyatlov | Mathematical theory of scattering resonances[END_REF], scattering resonances have a direct relation to quantum mechanics. They arise in the study of scattering of waves and particles. Physicists learned a lot about the forces and interactions in atoms and nuclei from scattering experiments. Atoms are bombarded with beams of particles. Then these particles are detected by devices that give the intensity as a function of the scattering angle or energy.

The positive Laplacian -∆ on L 2 (R d ) represents in quantum mechanics the kinetic energy operator of a free quantum particle. The many interactions, for instance which occurs in a scattering experiment, are represented by a real-valued function, called

potential V ∈ L ∞ c (R d ).
We are assuming without loss of generality that V has compact support, because at infinity the interactions with the scattering center are zero. The energy operator in this case is nothing else but a Schrödinger operator, also called perturbed Hamiltonian (perturbed by V )

H V = -∆ + V . ( 13
) By the Kato-Rellich Theorem, H V is self-adjoint. Hence its spectrum is real. Its resolvent R(λ 2 ) = (H V -λ 2 ) -1
is well defined on the upper half plane (λ) > 0 as a bounded operator on L 2 (R).

When restricted to C ∞ c (R), R(λ 2 ) extends to a meromorphic function of λ on the complex plane. The singularities of the extended resolvent are called the resonances of this Schrödinger operator. In general, it's very difficult to compute them. It is possible in very special cases, and for very special potentials. A well known case correspond to the case of the Laplace operator on R n (see section II.1.1). The goal of this thesis is to do the same computations, replacing R n by a homogeneous vector bundle (on a symmetric space of non-compact type and real rank one).

Let us do a more concrete computation related to the wave equation. In fact, the solution of this equation has a rate of oscillation and decay given by the scattering resonances. In dimension one, the wave equation is given by

(∂ 2 t + H V ) u(t, x) = (∂ 2 t -∂ 2 x + V (x)) u(t, x) = 0 . Assume the initial conditions u(0, x) = u 0 (x) and ∂ t w(0, x) = w 1 (x), where w j ∈ C ∞ c (R).
Then for any A > 0, we have the resonance expansion (Fourier series in the case where V = 0)

w(t, x) = λ j >-A e -ilλx f λ,l (x) + E A (t, x) ,
where the λ j vary over the set of the poles of the extended resolvent R(λ), the sum is finite, (H V -λ 2 ) l+1 f λ,l = 0 and the error term E A (t, x) is bounded by e -At in a well defined sense, see [DZ19, Theorem 2.9].

The poles λ j with (λ j ) < 0 of the extended resolvent are called scattering resonances or just resonances of the operator H V . Numerical simulations of the solutions show these functions have peaks decreasing at infinity. One remarks that the real part of the resonances is the rate of oscillations and the imaginary part is the opposite of the rate of the decay. The resulting rapidly growing theory attracts many researchers.

Of course, this is a brief overlook of the utility of the resonances in one physical setting. For applications to different situations we refer to [DZ19, Chapter 1].

Resonances of the Laplace operator on noncompact Riemannian symmetric spaces

Riemannian symmetric spaces of the non-compact type are important geometrical settings to study resonances of the Laplacian. Besides being intrinsically interesting objects, they play the role of model spaces to understand phenomena on more complicated or less regular geometries. In a typical setting, one works on a complete Riemannian manifold X with a finite geometry, for which the positive Laplacian ∆ is an essentially self-adjoint operator on the Hilbert space L 2 (X) of square integrable functions on X.

We suppose that ∆ has a continuous spectrum [ρ X , +∞[, with ρ X ≥ 0. The spectrum of ∆ might have some discrete parts, but these parts do not play any significant role, so we neglect them. Also, for simplicity, we assume to have shifted the Laplacian so that its spectrum has bottom at 0, and have changed variables z → z 2 for the resolvent so that, as above, the resolvent is analytic away from the real axis. The resolvent R(z) = (∆ -ρ X -z 2 ) -1 of the shifted Laplacian ∆ -ρ X is then a holomorphic function of z on the upper (and on the lower) complex half plane. For each such z, R(z) is a bounded linear operator from L 2 (X) to itself. As such, it cannot be extended across the real axis. However, let us restrict the resolvent to the dense subspace C ∞ c (X) of compactly supported smooth functions on X. Then the map z → R(z) might admit a meromorphic extension across the real axis to a larger domain in C or to a cover of such a domain. The poles, if they exist, are the resonances, also called quantum resonances or scattering poles, of ∆.

Let us introduce some notations. A Riemannian symmetric space of the noncompact type is a homogeneous space of the form X = G/K where G is a connected non-compact real semisimple Lie group with finite center and K is a maximal compact subgroup of G. The basic examples are the n-dimensional real hyperbolic spaces H n . In this case, the Lie group G is the Lorentz group SO e (n, 1) and K = SO(n). A Riemannian symmetric space of the non-compact type X has maximal flat subspaces, all of the same dimension, called the (real) rank of X. For instance, the rank of H n is 1. Since X is a symmetric space of the Lie group G, all natural operators acting on X, like the Laplacian and its resolvent, are G-invariant. They can therefore be studied using the representation theory of G. The analytic study of the resolvent of the Laplacian acting on functions on H n , in particular its meromorphic continuation across its spectrum, is classical and well-understood. It plays a central role when studying the resolvent on more general complete Riemannian manifolds for which the hyperbolic spaces are models. Still in the case of functions on a more general Riemannian symmetric space X = G/K, the study of resonances can in principle be done using an adapted harmonic analysis, the so called Helgason-Fourier analysis, which provides a diagonalization of the Laplacian and hence an explicit formula for its resolvent as a singular integral operator over the spectrum. This formula allowed Hilgert and Pasquale [START_REF] Hilgert | Resonances and residue operators for symmetric spaces of rank one[END_REF] to determine and study the resonances for an arbitrary X of rank one. The rank-one case was previously solved by Miatello and Will [START_REF] Miatello | The residues of the resolvent on Damek-Ricci spaces[END_REF] with different methods, in the context of Damek-Ricci spaces. The general higher-rank case is still open. Relevant works in this context are [START_REF] Mazzeo | Analytic continuation of the resolvent of the Laplacian on symmetric spaces of noncompact type[END_REF] and [START_REF] Strohmaier | Analytic continuation of resolvent kernels on noncompact symmetric spaces[END_REF]. Complete answers to the basic problems concerning the existence and location of the resonances of the Laplacian, as well as the representation-theoretic interpretation of the so-called residue operators at the resonances, are available only for (most of the) Riemannian symmetric spaces of rank 2. These results appeared in joint articles by Hilgert, Pasquale and Przebinda (see [START_REF] Hilgert | Resonances for the laplacian on products of two rank one riemannian symmetric spaces[END_REF][START_REF] Hilgert | Resonances for the Laplacian on Riemannian symmetric spaces: the case of SL(3, R)/SO(3)[END_REF][START_REF] Hilgert | Resonances for the Laplacian: the cases BC 2 and C 2 (except SO 0 (p, 2) with p > 2 odd)[END_REF]).

All the articles mentioned above consider the Laplacian acting on scalar functions on X. A more general question is to consider the Laplacian acting on sections of a homogeneous vector bundle on X. Such a bundle is determined by a finite-dimensional representation τ of K. Let us denote this bundle by E τ . The sections of E τ can be seen as vector-valued functions on G, with values in the space of the representation τ , such that

f (xk) = τ (k -1 )f (x) for all x ∈ G and k ∈ K .
The space of such functions which are smooth and compactly supported is denoted by C ∞ c (G, τ ). This means that we are replacing complex-valued functions on X with vector-valued functions which have specific transformation properties on the orbits of the compact subgroup K. This is a generalisation of the previous study, in the sense that the section of the homogeneous vector bundle E triv K , for the trvial representation of K, are exactly the complex valued functions on X.

Examples of sections of homogeneous vector bundles on X are the differential forms, the vector fields, and more generally the tensor fields on X: all these objects naturally arise in physical models, and it is therefore natural to look for resonances of the Laplacian in these settings.

The resolvent of the Laplacian of forms on a rank-one Riemannian symmetric space of the non-compact type has been studied by several authors; see [Cam97a, Cam05b, Cam05a, Cam97b, Ped94, CP04, BO00]. In particular, [START_REF] Carron | On the differential form spectrum of hyperbolic manifolds[END_REF] gives (for the differential forms on rank 1) the list of resonances and the Riemann surface on which the resolvent admits meromorphic extension. Nevertheless, to our knowledge, there is only one article studying the resonances and the residue operators of the Laplacian acting on the sections of a homogeneous vector bundle over X, namely [START_REF] Will | The meromorphic continuation of the resolvent of the Laplacian on line bundles over CH(n)[END_REF], where X is a complex hyperbolic space and the fibers have dimension one.

Results of the thesis

The goal of this thesis is to study the resonances of the Laplacian acting on the sections of a homogeneous vector bundle over a Riemannian symmetric space of the noncompact type X = G/K. The symmetric space is assumed to have rank one but the representation of K is arbitrary. Since every finite-dimensional representation of K decomposes into irreducibles, we restrict our attention to irreducible representations. As in the case of functions, the basic problems are to determine the existence, the localisation of the resonances and to study the residue operators associated with them.

This thesis is organized as follows. In section I, we introduce the notation and recall some basic facts about the structure of Riemannian symmetric spaces of the non-compact type and real rank one. There are four cases, listed in the following table:

G K X = G/K Spin(n, 1) Spin(n) real hyperbolic space SU(n, 1) S(U(n) × U(1)) complex hyperbolic space Sp(n, 1) Sp(n) quaternionic hyperbolic space F 4
Spin(9) octonion hyperbolic space

We set a to be a maximal flat subspace in p = T eK (X) the tangent space of X at the base point eK, and M the centralizer of a in K. In section I.9, we recall some facts on the generalisation of the Helgason-Fourier transform to homogeneous vector bundles. They are principally due to Camporesi [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF]. In particular, the Plancherel Theorem for L 2 -sections of the homogeneous vector bundles is given there. Denote the decomposition of τ over M as follows:

τ = σ∈ M (τ ) d σ σ (14)
where M (τ ) is the set of irreducible unitary representations of M which occurs in the restriction of τ to M , d σ is the degree of σ. We need some properties of the generalised spherical functions ϕ σ,λ τ associated with the irreducible representations σ ∈ M (τ ), which is detailed in section I.9. The explicit formula for the Plancherel density p σ corresponding to these σ ∈ M is given in Proposition II.1.4 (see also Appendix A). Corollary I.9.1 proves the convergence of the singular integral operator providing an explicit formula for the resolvent R of the Laplacian using the inversion formula of vector-valued Helgason-Fourier transform. In section II.1, we compute the resonances, which is the first main goal of this thesis. The holomorphic function z → R(z) is meromorphically extended from the complex upper half-plane to the whole space, using the residue theorem. The extended resolvent is a meromorphic function with simple poles on the imaginary axis: these poles are the resonances. This leads to our first theorem:

Theorem 1 Let G be a connected non-compact semisimple Lie group with finite center and with

Iwasawa decomposition G = KAN , where K be a fixed maximal compact subgroup of G. Suppose dim A = 1. Let M denote the centralizer of A in K. Let (τ, V τ ) be an irreducible unitary representation of K, and let E τ be the homogeneous vector bundle over G associated with τ . For each σ ∈ M (τ ), let N σ be the set of k ∈ Z such that

λ k := -i(B max + k)
is a pole of the Plancherel density (see (II.15) for the formula) and B max + k ≥ 0. Here B max is a nonnegative constant depending only on G and σ. We refer to (II.17), (II.18) and (II.20) for the precise definition.

In this setting, the meromorphic continuation of the resolvent R of the Laplace operator acting on the smooth compactly supported sections of E τ can be written as the sum R = σ∈ M (τ )

d σ R σ , ( 15
)
where R σ is given for all f ∈ C ∞ c (G, τ ) and for all N ∈ N by the following formula:

(R σ (ζ σ )f ) (x) = 1 |α| R-i(N+1/4) 1 ζ σ -λ|α| ϕ σ,λα τ * f (x) p σ (λα) λ dλ + 2iπ |α| k∈Nσ λ k >-i(N +1/4) 1 ζ σ -λ k |α| ϕ σ,λ k α τ * f (x) Res λ=λ k p σ (λα) λ (16)
In (16), α is the longest restricted root, ϕ λ,σ τ is a spherical function of type σ and

ζ σ := -z -ρ, ρ + µ σ + ρ M , µ σ + ρ M ) (17
S σ = (z, ζ) ∈ C 2 | ζ 2 := -z -ρ, ρ + µ σ + ρ M , µ σ + ρ M ) .
Then the resolvent R σ extends meromorphically from

S + σ = {(z, ζ) ∈ S | (ζ) > 0} to S σ .

The (simple) poles of this extension are the pairs

(z σ,k , ζ σ,k ) = (B max + k) 2 |α| 2 -ρ 2 α |α| 2 + µ σ + ρ M , µ σ + ρ M , -i(B max + k) 2 |α| 2 (18)
where µ σ is the highest weight of the representation σ, the numbers k are in N σ and ρ M is half sum of roots for M .

We refer to section I for the definitions of the various objects appearing in this theorem and to section II.1 for its proof.

The second problem we address in this article is the representation theoretic interpretation of the resonances. More precisely, consider the residual part of the meromorphic continuation of the resolvent in (16). For each pole λ k of the Plancherel density for σ ∈ M (τ ), one can introduce an operator, called the residue operator at λ k α, defined as follows:

R σ k : C ∞ c (G, τ ) -→ C ∞ (G, τ ) f -→ ϕ σ,λ k α τ * f (19)
Since the convolution product is on the left, it seems that it does not commute with the left translation of f . But it does, as we shall see in (I.22). As G acts on the image of R σ k by the left translations, we get a representation of G, called the residue representation at λ k α.

In section II.2 we restrict our attention to the representations τ which contains the trivial representation of M . In this case the structure of the principal series is well known [START_REF] Howe | Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations[END_REF][START_REF] Johnson | Composition series and intertwining operators for the spherical principal series[END_REF][START_REF] Johnson | Composition series and intertwining operators for the spherical principal series[END_REF]. The complexity of the general case (see [START_REF] David | Representations of rank one Lie groups[END_REF]) is formidable and might lead to much less pleasing results, thus we avoid it. We consider the family of resonances corresponding to σ = triv. To simplify the notation, we write R k instead of R triv k . Let E k be the residue representation at λ k α. We show that the E k 's are irreducible and equivalent to a subquotient of a spherical principal series representation of G. We determine which of them are unitary and which are finite-dimensional. Also, we identify their Langlands parameters and compute their wave front sets. The Langlands parameters are of the form (M A, δ, ν) and denotes the induced representation Ind G M AN (δ ⊗ e ν ⊗ triv). A lowest K-type of the induced representation with highest weight µ min identifies a unique irreducible subquotient of that induced representation (See [START_REF] Vogan | Classification of the irreducible representations of semisimple Lie groups[END_REF]). The wave front set of a representation has been introduced by Howe (see [START_REF] Howe | Wave front sets of representations of Lie groups[END_REF]). When G is semisimple, it is a closed set consisting of nilpotents orbits. For E k it turns out to be the closure of a single nilpotent orbit. In the following theorem, α is the longest restricted root as in Theorem 1 and for each restricted root β, the corresponding root space in g is denoted by g β . The minimal K-type of E k is given in the proof of the theorem in each case: they can be found in tables II.1, II.2, II.3 and II.4 respectively for the real, complex, quaternionic and octonionic hyperbolic spaces.

Theorem 2

Suppose that the representation τ contains the trivial representation of M . The residue representations E k are then irreducible.

1. If G = Spin(2n, 1), then τ has highest weight of the form (N, 0, . . . , 0), where N is a nonnegative integer.

• If N ≥ k+1, then E k has Langlands parameters M A, H k+1 (R 2n-1
), n -3 2 α with (k+1, 0, . . . , 0) as a lowest K-type's highest weight. Here H k+1 (R 2n-1 ) are harmonics of degree k + 1 on R 2n-1 . This representation is unitary. Its wave front set is the nilpotent orbit generated by g α .

• If N < k + 1, then E k has Langlands parameter M A, triv, (ρ α + k)α with the trivial representation as a lowest K-type. It is finite-dimensional. Also, it is non-unitary if k = 0. 2. If G = SU(n, 1), then τ has highest weight of the form (M 1 , 0, . . . , 0, -M 2 , -L),
where M 1 and M 2 are positive integers such that M 1 ≥ M 2 ≥ 0, L ∈ Z and

M 1 + M 2 + L is even. • If M 1 + M 2 ≥ 2k + 2 and |L| ≤ -2k -2 + M 1 + M 2 , then E k is unitary.
if n > 2, then E k has minimal K-type of highest weight (k+1), 0, . . . , 0, -(k+ 1), 0). Its Langlands parameter is M A, δ, n 2 -1 α where the highest weight of δ is (k + 1), 0, . . . , 0, -(k + 1), 0 . Its wave front set is the nilpotent orbit generated by g α/2 .

if n = 2, this representation is the discrete series with Blattner parameter ((k + 1), -(k + 1), 0, . . . , 0). Its wave front set is the nilpotent orbit generated by g α/2 .

• If L ≥ |-2k -1+M 1 +M 2 |+1, then E k is the representation with Langlands parameter M A, δ, k 2 + n 2 -1 2 α
, where the highest weight of δ is 0, 0, . . . , 0, -(k + 1), (k + 1)/2 . This representation is non-unitary. Its wave front set is the nilpotent orbit generated by the element n 2 of g α (see Lemma III.4.2 for the definition).

• If L ≤ |-2k -1+M 1 +M 2 |-1, then E k is the representation with Langlands parameter M A, δ, k 2 + n 2 -1 2 α
, where the highest weight of δ is (k + 1), 0, . . . , 0, (k + 1)/2 . This representation is non-unitary. Its wave front set is the nilpotent orbit generated by the element n 1 of g α (see Lemma III.4.2 for the definition).

• If M 1 + M 2 ∈ [0, 2k + 2[ and L < |2k + 2 -M 1 + M 2 |, then E k is the representation with Langlands parameter M A, triv, (ρ α + k)α . It is finite- dimensional and non unitary (if k = 0). 3. If G = Sp(n, 1), then τ has a highest weight of the form (t 1 , t 2 , 0, . . . , 0, t n+1 ),
where t 1 , t 2 and t n+1 are positive integers such that t 1 ≥ t 2 , t n+1 ≤ t 1 + t 2 and t 1 + t 2 + t n+1 is even.

• If t n+1 ≤ t 1 + t 2 -2k -4
, then E k is the representation with Langlands parameter M A, δ, ± n -3 2 α with τ as a lowest K-type, where the highest weight of δ is (k + 2, k + 2, 0, . . . , 0). This representation is non-unitary. Its wave front set is the nilpotent orbit generated by g α/2 .

• If t n+1 ≥ |t 1 + t 2 -2k -2|
the residue representation is unitary. Its wave front set is the nilpotent orbit generated by g α .

-If k ≤ 2n -4, then E k is the representation with Langlands parameter M A, δ, ± n -k 2 α with lowest K-type (k + 1, 0, . . . , 0, k + 1), where the highest weight of δ is (k + 1, 0, . . . , 0, k+1 2 ). -If k ≥ 2n -3, then E k is the discrete series representation with Blattner parameter µ k = (k + 1, 0, . . . , 0, k + 1).

• If t 1 + t 2 < 2k + 2 -t 1 -t 2 ,
• If b ≤ a -2k -8 , then E k is the representation with Langlands parameter M A, δ, 1 2 (k + 1) α
, where the highest of δ is k+4 4 (3, 1, 1, 1). This representation is non-unitary. Its wave front set is the nilpotent orbit generated by

g α/2 . • If b > a -2k -8 and b ≥ 2k + 2 -a, then E k is the representation with Langlands parameter M A, δ, 1 2 (k + 10) α , where the highest of δ is k+1 4 (3, 1, 1, 1
). This representation is unitary. Its wave front set is the nilpotent orbit generated by g α .

• If b < 2k + 2, then E k is the representation with Langlands parameter M A, triv, (ρ α + k)α . It is finite-dimensional and non unitary (if k = 0).

A nice consequence of our case-by-case results is the following.

Corollary 4.0.1

For a fixed k ∈ N, there is one-to-one correspondence between the irreducible subquotients of H λ k α and the (real) nilpotent orbits of g under the adjoint action. This correspondence maps each subquotient into the orbit whose closure is the wave front set of that subquotient. As we showed, E k is equivalent to one of these subquotients. Its wave front set is then the closure of one nilpotent orbit in g.

I Preliminaries

We shall use the standard notations N, Z, R, C and C × for the nonnegative integers, the integers, the real numbers, the complex numbers and the nonzero complex numbers.

For a complex number z ∈ C, we denote by (z) and (z) its real and imaginary parts.

The positive constants in the Haar measures do not matter in our computations and equalities. Integrals have to be considered up to positive multiples. We will denote by uppercase letters G, K, A, M, N Lie groups and by gothic letters their respecting Lie algebras g, k, a, m, n.

In this section we fix some notations and we recall some basic facts.

I.1 Context

Let G be a connected non-compact real semisimple Lie group with finite center. We recall that a Lie group is called semisimple if it is a direct product of simple groups (Its Lie algebra is nonabelian and contains no nonzero proper ideals). Let B(•, •) be the Killing form on the Lie algebra g of G, that is a bilinear form B :

g × g → C defined for all X, Y ∈ g by B(X, Y ) = Tr ad(X) • ad(Y ) .
Here ad is the adjoint action and can be defined through the Lie bracket

ad(X)(Y ) = [X, Y ] ,
for all X, Y ∈ g. We recall that for a semisimple Lie algebra, the Killing form is nondegenerate. An involution on g is a Lie algebra automorphism whose square is equal to the identity. Such an involution θ is called a Cartan involution on if B θ (X, Y ) := -B(X, θY ) is a positive definite bilinear form. We denote by k the set of fixed points of θ and by p the eigenspace of θ for the eigenvalue -1. In other words:

k = {X ∈ g | θX = X} and p = {X ∈ g | θX = -X} .
Then k is a Lie subalgebra of g. The corresponding connected Lie subgroup of G is maximal compact. We denote it by K. The Cartan decomposition of the Lie algebra g is given by: g = k ⊕ p .

We recall that here p is not a Lie algebra, but we have the following inclusions:

[k, p] ⊂ p , [k, k] ⊂ k , [p, p] ∈ k .
Let a be a maximal abelian subspace of p and A = exp a its associated subgroup of G. The exponential map exp : g → G restricts to a diffeomorphism between a and A. The inverse map is the logarithm denoted "log". The group A can also be seen as the maximal torus in exp p. The Cartan decomposition thus implies G = KAK, because p = k∈K Ad(k)a.

I.2 Roots and restricted roots systems

Let a * be the vector space of linear forms on a and a * C its complexification. The set Σ of restricted roots of the pair (g, a) consists of all linear forms α ∈ a * for which the vector space Let M be the centralizer of a in K, m its Lie algebra. We recall that a Cartan subalgebra h of a complex Lie algebra is a nilpotent algebra for which if [X, Y ] ∈ h for all X ∈ h, then Y ∈ h. Let t be a Cartan subalgebra of m. Then the Lie algebra h = t ⊕ a is a Cartan subalgebra of g. We denote by h C its complexification. The set Π of roots of the pair (g C , h C ) consists of all linear forms ε ∈ h * C for which the vector space

g α := {X ∈ g | [H, X] = α(H)X ,
g C,ε := {X ∈ g C | [H, X] = ε(H)X , for every H ∈ h C }
contains nonzero elements. These vector spaces are one-dimensional. We choose a set Π + of positive roots in Π which is compatible with Σ + , i.e. such that a root ε ∈ Π is positive when ε| a ∈ Σ + . Let also Π k (Π k + ) be the set of (positive) roots of the pair (k C , h C ).

We have the decomposition

g C = h C ε∈Π g C,ε .

I.3 The rank-one case

In this thesis, we are restricting ourself to real rank-one groups G. In other words, we suppose that a is one-dimensional and then isomorphic to R.

Rank-one symmetric spaces of the non-compact type are classified into three infinite families -namely, the real, complex and quaternionic hyperbolic spaces -and one exceptional example, the octonionic hyperbolic plane.

Since G is of real rank one, the set Σ is either equal to {±α} or {±α, ±α/2}. Among the groups listed in the table in the introduction, only G = Spin(n, 1) has restricted root system {±α}. As a system of positive roots Σ + we choose {α} and {α, α/2}.

Then ρ = 1 2 m α + m α/2 2
α, where we set m α/2 = 0, if Σ = {±α}. The Killing form B is positive definite on p, so X, Y := B(X, Y ) defines an Euclidean structure on p and on a ⊂ p. For all λ ∈ a * , let H λ denote the unique element in a such that H λ , H = λ(H) for all H ∈ a. We extend the inner product to a * by setting λ, µ := H λ , H µ for all λ, µ ∈ a * . Further, we denote the C-bilinear extension of •, • on a to a * C by the same symbol. We identify a * C to C by means of the isomorphism:

a * C -→ C λ -→ λ α := λ,α α,α (I.4) which identifies ρ with ρ α := 1 2 m α + m α/2 2
. We recall that the Lie algebra n = α∈Σ g α is nilpotent. More explicitly the property [g α , g β ] ⊂ g α+β for α, β ∈ Σ, allow us to see that g α/2 is nilpotent of degree 2 when it exists and g α is nilpotent of degree 1. It's important to consider the possible nilpotent orbits under Ad(G) in g which are needed in section III.

I.4 Homogeneous vector bundles

We recall that a vector bundle over a topological space X is pair (E, π) such that • E is a topological space such that π : E → X is continuous.

• For all x ∈ X, E x := π -1 (x) is a finite dimensional vector space. Its topology is that induced from E.

• For all x ∈ X, there exist a neighbourhood U of x in X, a topological vector space E 0 over R and α : π -1 (U ) → U × V , a homeomorphism such that:

-

p 1 • α = π where p 1 : U × E 0 → U ,
for all x ∈ U , the map α| Ex : E x → {x}×E 0 is an isomorphism of topological vector spaces.

The rank of the vector bundle is the dimension of the vector spaces E x .

A section of E is a smooth map from X to E such that the image of x is in the fiber E x .

Definition I.4.1

We say that E is a homogeneous vector bundle over G/K, if G acts on E by an smooth action such that:

1. g • E x = E gx , for all x ∈ G/K and g ∈ G, where gx is the action of G on G/K, 2. the map g• : E x → E gx , (x, v) → g • (x, v) is linear for all g ∈ G and x ∈ G/K
Every homogeneous vector bundle can be constructed as follows. We fix a finitedimensional unitary representation (τ, V τ ) of K. The group K acts on G × V τ on the right, such that for all k ∈ K, g ∈ G and v ∈ V τ ,

(g, v) • k = (g, v)k := (gk, τ (k -1 )v)
This action of K is a homeomorphism on G × V τ . We can then quotient the topological product space G × V τ by this action of K.

Let E τ = G × τ V τ := (G × V τ )/H and p : E τ → G/H such that p((g, v)H) = gH.
This map is well defined and surjective. Moreover, if we provide E τ and G/K with their respective quotient topology, p is continue. In these conditions, E τ can be equiped with a structure of homogeneous vector bundle. In fact, one can prove that every homogeneous vector bundle on X = G/K is isomorphic to E τ for a well choosen τ ∈ K.

Lemma I.4.1

Let E a homogeneous vector bundle on G/K. Let V τ := E 0 = E eK . Since G (and so K) acts on the fibers of E, we obtain a represention τ :

K → GL(E eK ) which is continuous. Let V τ := E 0 = E eK . Thus E is isomorphic to G × τ V τ = E τ .
For the proofs and more properties on homogeneous vector bundles, we refer the reader to [Wal73, §5.2 p. 114].

So from now on, we fix a finite-dimensional unitary representation (τ, V τ ) of K. Let E τ := X × τ V τ denote the homogeneous vector bundle over X. We write Γ ∞ (E τ ) for the space of all smooth sections of E τ . As proved in [Wal73, §5.4 p. 119], there is an isomorphism of vector spaces between Γ ∞ (E τ ) and K,τ,τ ) are sometimes called the radial systems of sections of E τ . The link with the sections comes by the fact that for every v ∈ V τ , the function F (•)v is a smooth section of E τ .

C ∞ (G, τ ) := {f : G smooth -→ V τ | f (xk) = τ (k -1 )f (x) for all x ∈ G and k ∈ K} Set C ∞ (G, K, τ, τ ) := {F : G smooth -→ End(V τ ) | F (k 1 xk 2 ) = τ (k -1 2 )F (x)τ (k -1 1 )} where x ∈ G and k 1 , k 2 ∈ K. The elements F ∈ C ∞ (G,

Remark

The three following algebras are * -isomorphic (as C * -algebras, see [vD69]) :

• The algebra C ∞ (G, K, τ, τ ). The involution is given by f * (x) = f (x -1 )
* where the star in the right-hand side is the Hilbert space adjoint.

• The algebra

C ∞ χ (G) := {f : G smooth -→ C | χ * f * χ = f and f (k -1 xk) = f (x) ∀x ∈ G and k ∈ K} ,
where for all k ∈ K, χ(k) := d τ Tr τ (k -1 ). The involution is given by f * (x) = f (x -1 ).

• For a fixed

i ∈ [[1, d]]. The algebra C ∞ e i (G) := {f : G → C smooth | e i * f * e i = f }
where for all k ∈ K, e i (k) := d τ τ ii (k -1 ). The involution is given by f * (x) = f (x -1 ).

The "last" algebra is particularly interesting, because for each i the algebras are *isomorphic. This is quite surprising at a first look. One could choose any vector in v ∈ V τ and have

C ∞ v (G) C ∞ (G, K, τ, τ ).

Remark (When

τ is not irreducible) If τ is not irreducible then E τ = i E τ i
, where τ i are the irreducible components of τ .

Studying the sections of E τ amounts to studying the sections of each bundle E τ i . We can therefore suppose without loss of generality that τ is irreducible. We notice that if τ is the trivial representation "triv" of K on the one-dimensional vector space C. Then the sections of E triv are the functions on G/K. They can be seen as right-K-invariant functions on G. Moreover, in this case, the radial systems of sections agree with the K-bi-invariant functions on G. We will refer to this situation as the scalar case.

I.5 Principal series representations

Let M be the set of all equivalence classes of irreducible unitary representations of M . For (σ, H σ ) ∈ M and λ ∈ a * C , we denote by

τ λ = π σ λ = Ind G M AN (σ ⊗ e iλ ⊗ triv)
the induced representation from M AN to G by the representation σ ⊗ e iλ ⊗ triv. We will use the same notation for its derived representation of g too. The representation space H σ λ of π σ λ is the Hilbert space completion of

{f : G → H σ | f (xman) = a -iλ-ρ σ(m -1 )f (x) for all x ∈ G, m ∈ M, a ∈ A and n ∈ N } (I.5) with respect of the L 2 -inner product f, g σ = K f (k), g(k) Hσ dk ,
where •, • Hσ is an inner product on H σ making σ unitary. The action of π σ λ on H σ λ is given by

π σ λ (g)f (x) := f (g -1 x) for all g, x ∈ G and f ∈ H σ λ . The set {π σ λ | λ ∈ a * C , σ ∈ M } is called the minimal principal series of G. If σ = triv M then it is called the spherical principal series.
The compact picture of the principal series representations is obtained by restriction of the elements of H σ λ to K. Its representation space, which we denote by H σ , is the Hilbert completion of:

{f : K → H σ | f (km) = σ(m -1 )f (k) for all k ∈ K, m ∈ M }
with respect to L 2 inner product. It is independent of λ. The action is given by:

π σ λ (g)f (k) := e -(iλ+ρ)H(g -1 k) f (k(g -1 k))
for all g ∈ G, k ∈ K and f ∈ H σ . The representation π σ λ is unitary for λ ∈ a * .

In the above definitions, we are following the notation of [Hel08, chapter VI, §3] and [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF]. In the literature (for instance in [START_REF] Vogan | Representations of real reductive Lie groups[END_REF]), one often finds (π σ iλ , H σ iλ ), instead of our (π σ λ , H σ λ ) for the representation Ind G M AN (σ ⊗ e iλ ⊗ triv).

In the following, when working with principal series, we actually work with their Harish-Chandra modules (see the next paragraph I.6 for the definition). The restriction of π σ λ to K is the representation Ind K M σ of K induced from σ. In particular, because of Frobenius reciprocity theorem, for any τ ∈ K:

m(τ, π σ λ | K ) = m(σ, τ | M ) .
Here the symbol m(β, α) denotes the multiplicity of the irreducible representation β in the representation α. We say that τ is a K-type of π σ λ if it occurs in π σ λ | K . We say that τ is a minimal K-type of an admissible representation π of G if and only if its highest weight µ minimizes the Vogan norm

µ V = µ + 2ρ K , µ + 2ρ K in the set of K-types of π. Here 2ρ K is the sum of positive roots of the pair (k C , h C | k C ).
[Vog77, Theorem 1] ensures that each minimal K-type τ min has multiplicity one in π σ λ . Therefore, if we fix this τ min there exists a unique irreducible subquotient J(σ, λ, µ) of π σ λ containing τ min . These representations are so important because they appear in the continuous part of the Plancherel formula. Let G acts on L 2 (G) by left translation:

g • f (x) = f (g -1 x) , for all g, x ∈ G and f ∈ L 2 (G) .
Harish-Chandra proved the decomposition of L 2 (G) for any reductive group. In the rank one case, this can be written:

L 2 (G) = Ĝ H π ⊗ H * π dν(π) = σ∈ M ia * H σ λ ⊗ H σ λ * dν(λ) ⊕ ω d ω H ω ⊗ H * ω , (I.6)
where dν is the Plancherel measure and the last direct sum is over the discrete series ω of G. For recall π is a discrete series of G if there exists a nonzero matrix coefficient of π which belongs to L 2 (G). The matrix coefficients of a representation π are the functions π(•)u, v from G to C, where u, v are in the Hermitian space V π of the representation π and •, • is a Hermitian product on V π . This is equivalent to ask that all matrix coefficients of π belong to L 2 (G). If (π, V π ) is a discrete series then there exist

d π ∈ R + , called the formal degree of π, such that G π(g)u, v π(g)u , v dg = d -1 π u, v u , v , with u, u , v, v ∈ V π . (I.7)
In the rank one case, there is always discrete series, unless if g = so(2n + 1, 1). This is also the only case, when there are no resonances of the Laplace operator.

The abstract Plancherel formula (I.6) for L 2 (G, τ ) can be then written as follows (see [BOS94, page 4]):

L 2 (G, τ ) = σ∈ M ia * H σ λ ⊗ (H σ λ * ⊗ V τ ) K dν(σ, λ) ⊕ ω d ω H ω ⊗ (H * ω ⊗ V τ ) K , (I.8)
where K denotes the K-invariant vectors. In other words, (H * ⊗V τ ) K = Hom K (H , V τ ), that is to say the space of intertwining operators between H | K and V τ .

I.6 Langlands parameters of a representation

The irreducible admissible representations of the group G can be classified by their Langlands parameter. Let us recall some facts and definitions.

A representation V of G is said admissible, if its restriction to

K V | K = γ∈ K V (γ)
is such that dim V (γ) is finite. Every unitary and/or irreducible representation is admissible.

We say that an admissible representation (π,

V ) is tempered if all K-finite matrix coefficients π(•)u, v , with u, v ∈ V are in L p (G) for any p > 2. A G-module V is a (g, K)-module if: • For all v ∈ V , k ∈ K and X ∈ g, k • (X • v) = (Ad(k)X) • k • v,
• For all v ∈ V , K • v spans a finite dimensional subspace of V on which K acts continuously,

• For any v ∈ V , Y ∈ k, d dt exp(tY ) • v t=0 = Y • v.
We say that a (g, K)-module V is a Harish-Chandra module if it is admissible and finite dimensional as a U (g)-module.

Theorem I.6.1 (Casselman subrepresentation theorem) Every Harish-Chandra module can be embedded in a principal series representation.

However it's not clear how it is embedded.

The group M AN is called a minimal parabolic subgroup of G. The others minimal parabolic subgroups are the conjugates of M AN . A group P is called parabolic if it is closed and if it contains a minimal parabolic subgroup. We denote by s its Lie algebra. There exists a canonical decomposition of s, called Langlands decomposition s = m P ⊕ a P ⊕ n P , such that:

• m P , a P and n P three orthogonal Lie subalgebras of g with respect to the inner product

B(•, θ•) , • m P ⊕ a P = s ∪ θ(s), • a = p ∩ Z m P ⊕a P
If we define M 0,P , A P and N P as the connected Lie subgroups with Lie algebras m P , a P and n P respectively, and M P = Z K (a)M 0,P . Then

P = M P A P N P
is the Langlands decomposition of the group P .

Theorem I.6.2 ([Kna01, Theorem 7.24'])

Let P = M P A P N P be a parabolic subgroup of G. Let σ be an irreducible tempered representation of M and let ν ∈ a * such that (ν) is in the open positive Weyl chamber. Then Ind G M AN (σ ⊗ e ν ⊗ triv), as a Harish-Chandra module, has unique irreducible quotient J(P, σ, ν). J(P, σ, ν) is called the Langlands quotient of the representation of the principal series.

We are now able to cite the Theorem of Langlands parameters :

Theorem I.6.3 ([Kna01, Theorem 8.54]) Fix a minimal parabolic subgroup M AN of G. There is a one to one correspondence between the equivalence classes of irreducible admissible representations π of G and the triples (P, [γ], ν) such that:

• P = M P A P N P is a parabolic subgroup containing M AN • γ is an irreducible tempered representation of M and [γ] is its equivalence class • ν is a member of a * C .
The correspondence is that (P, [γ], ν) corresponds to the class of J(P, γ, ν). The data (P, [γ], ν) are called the Langlands parameters of the admissible representation π and π is equivalent as a Harish-Chandra module to J(P, γ, ν).

Another invariant of the representation π is its minimal K-type τ min . Suppose now that G is of real rank-one. Then up to conjugation there are just two parabolic subgroups containing the minimal parabolic subgroup M AN : G and M AN . Then for any irreducible admissible representation π of G, π belongs to the discrete series (when the parabolic subgroup is G) or there exists a pair (δ, ν) as follows: Let τ min be a minimal K-type in π. Then there exists δ such that τ min is also the minimal K-type in the induced representation of δ to K. The parameter ν is the element of a C such that π is infinitesimally equivalent to J(M AN, δ, ν) in the principal series Ind G M AN (δ ⊗e ν ⊗triv). This is the unique subquotient containing τ min as lowest K-type. For more explanations the reader can follow [Vog81, Sections 4.1, 4.2, 6.5, 6.6].

I.7 Homogeneous differential operators

We recall the definition of differential operator on vector bundles.

Definition I.7.1 (Differential operator)

Let M a manifold and E, F vector bundles over M of rank k and l respectively. we say that D is a differential operator of order m from E to F , if D : Γ ∞ E → Γ ∞ F is a linear map such that for all x ∈ M there exists a neighbourhood U of x in M with local coordinates x 1 , . . . , x n and local trivialisations, ψ :

E| U → U ×K k and φ : F | U → U ×K l of E and F respectively such that if f ∈ Γ ∞ E, y ∈ U , if (ψ • f )(y) = (y, S(y)) and if (φ • Df )(y) = (y, T (y)) then : T (y) = |I|≤m A I (y) ∂ |I| ∂x I S(y)
where

A I is a map C ∞ from U to Hom(K k , K l ).
A homogeneous differential operator D on E τ is a linear differential operator from Γ ∞ (E τ ) to itself which is invariant under the G-action by left translations, that is L(g)D = DL(g) for all g ∈ G .

(I.9)

The set of homogeneous differential operators on E τ is an algebra with respect to composition. We denote it by D(E τ ). It acts on C ∞ (G, τ ) because of the isomorphism with the space smooth sections Γ ∞ (E τ ). Unlike in the scalar case, i.e. when τ is the trivial representation, this algebra need not be commutative. Let U (g C ) be the universal enveloping algebra of the complexification g C of g. Each element of U (g C ) induces a left-invariant differential operator on G by:

X 1 • • • X k • f (g) := ∂ ∂t 1 ∂ ∂t 2 • • • ∂ ∂t k f (g exp t 1 X 1 exp t 2 X 2 • • • exp t k X k ) t 1 =...=t k =0 (I.10) for all X = X 1 • • • X n ∈ U (g C ), f ∈ C ∞ (G) and g ∈ G.
Let U (g C ) K denote the subalgebra of the elements in U (g C ) which are invariant under the adjoint action Ad of K. The elements of U (g C ) K act on C ∞ (G, τ ) as homogeneous differential operators. Since K is compact, Theorem 1.3 in [START_REF] Minemura | Invariant differential operators and spherical sections on a homogeneous vector bundle[END_REF] ensures that all elements of D(E τ ) can be written as an element of U (g C ) K . But there is no isomorphism in general. More explicitly, there exists a map from U (g C ) K to D(E τ ) such that its kernel is U (g C )I, where I is the kernel of τ in U (h C ).

We can extend this action to the set of radial system of section C ∞ (G, K, τ, τ ) by setting:

D • φ v := D • (φ • v) for all D ∈ U (g C ), φ ∈ C ∞ (G, K, τ, τ ) and v ∈ V τ .

I.8 The Laplace operator

Let {X 1 , . . . , X dim g } be any basis of g. We denote by g ij the ij-th coefficient of the inverse of the matrix B(X i , X j )

1≤i,j≤dim g

, where B is the Killing form. The Casimir operator is defined by

Ω := 1≤i,j≤dim g g ij X j X i .
If X k k=1,...,dim k and X k k=dim k+1,...,dim g are, respectively, orthonormal bases of k and p with respect to B θ , then:

Ω = - dim k i=1 X 2 i + dim g i=dim k+1 X 2 i .
In fact, Ω is in the center of U (g C ). The invariant differential operator corresponding

-Ω is the positive Laplacian ∆.

We can extend any representation of g to g C by linearity and to a representation of the associative algebra U (g C ). These representations will always be denoted by the same symbol. Since Ω is in the center of U (g C ), the linear operator π σ λ (Ω) is an interwining operator of the representation π σ λ for all λ ∈ a * C and σ ∈ M . Lemma 4.1.8 in [START_REF] Vogan | Representations of real reductive Lie groups[END_REF] ensures that π σ λ (Ω) acts by a scalar. To compute this scalar, one can use [Kna01, Proposition 8.22 and Lemma 12.28], and get that:

π σ λ (Ω) = -λ, λ -ρ, ρ + µ σ + ρ m , µ σ + ρ m Id . (I.11)
Here µ σ is the highest weight of σ and ρ m is half sum of the positive roots ε ∈ Π + such that ε| a = 0.

I.9 The vector-valued Helgason-Fourier transform and spherical functions of type τ

In this section we review some basic facts on Camporesi's extension of the Helgason-Fourier transform to homogeneous vector bundles. We refer the reader to [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF] for more information.

We keep the notations of the introduction. In particular, since we suppose that G is of real rank one, in the Plancherel formula only (minimal) principal series representations, and if G = Spin(2n + 1, 1), discrete series representations occur.

Let (τ, V τ ) be an irreducible unitary representation of K. Let

M (τ ) = {σ ∈ M | m(σ, τ | M ) ≥ 1}
denote the set of unitary irreducible representations of M which occur in the restriction of τ to M . We will denote by d γ the dimension of a representation γ. For σ ∈ M (τ ), let P σ be the projection of V τ onto the subspace of vectors of V τ which transform under M according to σ. Explicitly,

P σ = d σ M τ (m -1 )χ σ (m) dm , (I.12)
where χ σ denotes the character of σ. We denote by p σ (λ) the Plancherel density associated to the principal series representation π σ λ . Recall the Iwasawa decomposition (I.2) of x ∈ G. According to [Cam97a, Theorem 1.1], the vector-valued Helgason-Fourier transform of

f ∈ C ∞ c (G, τ ) is the function from a * C × K to V τ defined by f (λ, k) = G F iλ-ρ (x -1 k) * f (x) dx . (I.13)
Here, for µ ∈ a C and x ∈ G,

F µ (x) = e µ(H(x)) τ (k(x)) (I.14)
and * denotes the Hilbert space adjoint.

In the rank-one case, the inversion formula is

f (x) = 1 d τ σ∈ M (τ ) a * K F iλ-ρ (x -1 k) P σ f (λ, k) p σ (λ) dλ dk + γ∈D G C γ K F -iµ-ρ (x -1 k) P γ f (iµ, k) dk (I.15)
Here µ ∈ a * and γ ∈ M are chosen so that γ is infinitesimally equivalent to a subrepresentation of Ind G M AN (γ ⊗ e µ ⊗ triv). Moreover, C γ is a suitable constant depending on γ. The set D G is the set of discrete series of G. It consists of all irreducible unitary representations of G whose matrix coefficients are in L 2 (G). The second sum term is called the discrete part of the Plancherel formula. In the following, we will disregard this term. In fact, only the first term, i.e. the continuous part of the Plancherel formula, can contribute to the resonances, by means of the poles of the Plancherel density.

As a consequence not involving discrete series, Parseval's formula for the continuous spectrum reads as follows: for

f, h ∈ C ∞ c (G, τ ) f, h c = σ∈ M (τ ) 1 d σ a * K P σ f (λ, k), P σ h(λ, k) p σ (λ) dλ dk , (I.16)
where the index c underlines that we are only considering the contribution from the continuous spectrum. See [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF]p. 286]. On the right-end side of (I.16), •, • denotes the inner product on V τ making τ unitary. The corresponding norm will be denoted by u = u, u . We will also need a vector-valued analogue of Harish-Chandra's spherical functions on a non-compact reductive Lie group. These vector-valued functions were introduced by Godement [God52] and Harish-Chandra [START_REF] Harish-Chandra | Differential equations and semisimple Lie groups in Collected Papers III[END_REF]. They depend on the fixed representation τ of K and on a representation of the principal series indexed by σ ∈ M (τ ) and λ ∈ a * C .

Keep the above notation for the principal series. Let P τ denote the projection of H σ λ onto its subspace of vectors which transform under K according to τ , that is, Let Hom K (H σ λ , V τ ) be the space of K-intertwining operators between π σ λ | K and τ . We equip this space with the scalar product P, Q = 1 dτ Tr(P Q * ), where * denotes the adjoint. We fix an orthonormal basis {P ξ } ξ=1,...,m(σ,τ | M ) of this space. Then

P τ = d τ K π σ λ (k)χ τ (k -1 )
ϕ σ,λ τ (g) = m(σ,τ | M ) ξ=1 P ξ π σ λ (g) P * ξ . (I.20)
See pp. 268-269 and 273 in [START_REF] Camporesi | The Helgason Fourier transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF].

Lemma I.9.1

The spherical functions ϕ σ,λ τ are even functions of λ ∈ a * for all σ.

Proof. Due to Lemma 3.1 in [START_REF] Camporesi | The spherical transform for homogeneous vector bundles over Riemannian symmetric spaces[END_REF], the spherical functions ϕ σ,λ τ are in one-to-one correspondence with their traces. Now Tr(ϕ σ,λ τ ) = m(σ, τ | M ) χ τ * Θ σ λ , where χ τ and Θ σ λ are the respective characters of τ and π σ λ . Lemma 4, page 162, in [START_REF] Harish-Chandra | Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula[END_REF] gives us that Θ σ λ = Θ -σ -λ because -1 is in the Weyl group. As -1 acts trivially on M (so on σ), the lemma follows.

The spherical function ϕ σ,λ τ can be described as an Eisenstein integral (see [Cam97a, Lemma 3.2]),

ϕ σ,λ τ (x) = d τ d σ K τ (k(xk)) P σ τ (k -1 ) e (iλ-ρ)(H(xk)) dk . (I.21) Notice that ϕ σ,λ τ satisfies ϕ σ,λ τ (k 1 xk 2 ) = τ (k 1 )ϕ σ,λ τ (x)τ (k 2 ) for every x ∈ G and k 1 , k 2 ∈ K. The convolution with a function f ∈ C ∞ c (G, τ ) is defined by: (ϕ σ,λ τ * f )(x) := d τ d σ G ϕ σ,λ τ (x -1 g)f (g) dg . (I.22)
According to [Cam97a, Proposition 3.3], it can be expressed in terms of the vectorvalued Helgason-Fourier transform of f :

(ϕ σ,λ τ * f )(x) = d τ d σ K F iλ-ρ (x -1 k) P σ f (λ, k) dk . (I.23) Lemma I.9.2
The spherical functions ϕ σ,λ τ are joint eigenfunctions of the homogeneous differential operators on E τ . Moreover, for all z ∈ Z(g C ), the center of U (g C ), we have

z • ϕ σ,λ τ = γ(z)(iλ -µ σ -ρ m ) ϕ σ,λ τ . (I.24)
Here γ is the Harish-Chandra homomorphism described in [Kna01, Chapter VIII, paragraph 5] and µ σ is the highest weight of σ.

Proof. In fact, the function Ψ λ defined in [START_REF] Yang | Poisson transforms on vector bundles[END_REF] by

Ψ λ (nak) := τ (k -1 )a λ+ρ , with k ∈ K, a ∈ A, n ∈ N,
is nothing but the function x → F -λ-ρ (x -1 ) defined in (I.14). Hence [Yan98, Proposition 1.3, Corollary 1.4 and Theorem 1.6] allows us to prove (I.24).

Remark

For the Casimir operator, the eigenvalue is given by (I.11) :

γ(Ω)(iλ -µ σ -ρ m ) = -λ, λ -ρ, ρ + µ σ + ρ m , µ σ + ρ m (I.25)
To compute the resonances of the Laplacian, we need to describe the vector-valued Helgason-Fourier transform of functions on C ∞ c (G, τ ). By the Cartan decomposition, we can uniquely write an element x ∈ G as x = k exp X, with k ∈ K and X ∈ p. For X ∈ p, we write |X| = B(X, X) 1/2 . Define the open ball centered at 0 and of radius R > 0 in a ⊂ p by

B R a = {X ∈ a |X| < R} .

Moreover, we denote the geodesic distance between o = eK in xK by d(o, xK).

Let

B R = {x ∈ G d(o, xK) ≤ R} .
We fix an orthonormal basis {e 1 , . . . , e dτ } of V τ , then we define

|| f (λ, k)|| 2 := dτ i=1 f (λ, k), e i 2 .
The direct implication of the Paley-Wiener theorem for C ∞ c (G, τ ) is given by the following Lemma.

Lemma I.9.3 Let f be in C ∞ c (G, τ ) and R > 0. If supp f ⊂ B R , then f (λ, k) is an entire function of λ ∈ a * C for all N ∈ N: sup λ∈a * C ,k∈K e -R | (λ)| (1 + |λ|) N || f (λ, k)|| < ∞ ( * )
Proof. Using the integration formula with respect to the Iwasawa decomposition G = AN K (for example in [Hel00, Ch.I §5 Corollary 5.3]), one can prove that for all

f ∈ C ∞ c (G, τ ) f (λ, k) = F a L k -1 f (I.26)
Here λ ∈ a * and k ∈ K, f (g) := e ρ(H(g)) N f (gn)dn is called the Radon transform of f and

F a (φ)(λ) = a φ(X)e iλ(X) dX (I.27)
the Fourier transform on a for φ ∈ C ∞ c (a) and λ ∈ a * . Recall that the Euclidean Paley-Wiener theorem ensures that F a (φ) is an entire function of the exponential type and rapidly decreasing, i.e. if supp φ ⊂ B a R then

∀N ∈ N, ∃C N ≥ 0, so that |F a (φ)(λ)| ≤ C N (1 + |λ|) -N e R | (λ)| Define fi (•) := f (•), e i .
Then fi is a smooth function.

The idea of the proof is as follows:

supp f ⊂ B R 2 =⇒ supp L k -1 fi ⊂ B a R 1 ⇐⇒ (
e -r| λ| (1 + |λ|) N | f (λ, k), e i | < ∞ .
Moreover, by (I.26), this is also equivalent to

sup λ∈a * C , k∈K, i=1,...,dτ e -r| λ| (1 + |λ|) N |F A L k -1 fi (λ)| < ∞ .
In turn, by the Paley-Wiener theorem for the Fourier transform on a, this is equivalent to supp L k -1 fi ⊂ B a R as a function on a for every i and for every k ∈ K.

2 Suppose supp f ⊂ B R and let X ∈ B a R . For all k ∈ K and n ∈ N , due to [Hel00, Chapter IV, (13)]:

d(o, ke X nK) > H(ke X n) = |X| ≥ R So ke X n ∈ B R ⊃ supp f and then fi (ke X ) = 0 for every i which implies that X ∈ supp L k -1 fi .
From Lemma I.9.3 using (I.23) and the fact that K is compact, we obtain the following corollary.

Corollary I.9.1 For every function f ∈ C c (G, τ ), the convolution product ϕ σ,λ τ * f is an even entire function of λ ∈ a * C with the property that there exists a constant r > 0 such that for all N ∈ N the following inequality holds:

sup λ∈a * C e -r| λ| (1 + |λ|) N ||ϕ σ,λ τ * f || < ∞ . (I.28) Remark Campoli ([Cam80]
) proves the Paley-Wiener theorem for the case of functions f ∈ C ∞ c (G) in (real) rank one which are invariant by convolution on K with τ 's character χ τ := d τ Tr τ , i.e. f = χ τ * K f * K χ τ . We denote this algebra by C ∞ c,χτ (G). There is a well known-isomorphism between C ∞ c,χτ (G) and the space C ∞ c (G, K, τ, τ ) of radial systems of sections. It's given by Ξ :

C 0 c,χ (G) → C c (G, K, τ, τ ) defined for all f ∈ C 0 c,χ (G) by Ξ(f )(x) := K f (k -1 x)τ (k -1 )dk .
Its inverse is given for all Ξ(f )

∈ C c (G, K, τ, τ ) by Ξ(f ) → d τ Tr Ξ(f )
This yields a Paley-Wiener theorem for C ∞ c (G, K, τ, τ ). As pointed out in the previous section, there is a link between C c (G, K, τ, τ ) and C c (G, τ ), but no isomorphism.

A general Paley-Wiener theorem was proved by Arthur in a general setting, without restricting to the rank one case ([Art83, Th.

3.3]). It concerns functions

f : G → V τ such that f (k 1 xk 2 ) = τ (k 1 )f (x)τ (k 2 )
where τ is a two side representation of K.

Remark

One can prove the Paley-Wiener theorem for the compactly supported sections of the homogeneous vector bundle E τ following the proof of [BOS94, Lemma 2.3], modified so that it applies to the Helgason-Fourier transform of Camporesi. The theorem is cited as follows.

Let f be in C ∞ c (G, τ ) and r > 0. Consider the following conditions:

(1) supp f ⊂ B r (2) supp f ⊂ B a r (3) sup λ∈a * C ,k∈K e -r| (λ)| (1 + |λ|) N || f (λ, k)|| < ∞, for all N ∈ N Then (1) ⇔ (2)

and (1) ⇒ (3). If we suppose that G has one conjugacy class of Cartan subgroups then (3) ⇒ (1).

For the computations in this thesis we only need one direction of the Paley-Wiener type Theorem, i.e. Lemma I.9.3. The theorem above is built from a theorem of Delorme [START_REF] Delorme | Théorème de type Paley-Wiener pour les groupes de Lie semisimples réels avec une seule classe de conjugaison de sous groupes de Cartan[END_REF] which supposes that G has one conjugacy class of Cartan subgroups. In the rank one case, this theorem holds only for G = Spin(2n + 1, 1) (when there are no discrete series). Recall that in that case, there are no resonances because the Plancherel measure has no poles.

II Computation of the resonances and the residue representations

In this chapter, we prove Theorem 1, which lists the resonances of the Laplace operator on homogeneous vector bundles on rank-one Riemannian symmetric spaces of noncompact type. As preliminary examples, we present the cases of the Laplace operator on R n and on the homogeneous line bundles over H 2 (R) SL(2, R) SO(2) . These cases are known, but are complementary because they are not included in our general rank-one case treated in section II.1.3.

Each resonance produces a representation which we describe below. Let R be the resolvent of the Laplace operator. For each resonance z k , G acts by left translation on the space

{R(z k )f | f ∈ C ∞ c (G, τ )} .
This is a representation attached to z k , which we call in this thesis residue representation at z k . We show that all these representations are irreducible, find their Langlands parameters, and determine which of them are unitarizable.

II.1 Computation of the resonances

II.1.1 Example of R n Let X = R n , with n ≥ 2. The Laplace operator ∆ = - n j=1 ∂ 2 ∂x 2 j is self-adjoint on L 2 (R n ) and
z → R(z) ∈ Hom(C ∞ c (R n ), C ∞ c (R n ) ) , where C ∞ c (R n ) is the dual of C ∞ c (R n ).
C \ [0, +∞[, f ∈ C ∞ c (R n ) and x ∈ R n , (∆ -z) -1 f (x) = (2π) n R n f (λ)(|λ| 2 -z) -1 e iλ•x dλ . (II.2) Hence, R(z)f (x) = (2π) n R n f (λ)(|λ| 2 -z) -1 e iλx dλ = (2π) n R n R n f (y) e iλ(x-y) (|λ| 2 -z) -1

dy dλ

If n = 1 (which corresponds to the "rank-one case"), we change the variable

ζ = √ z, choosing √ -1 = i. Hence (ζ) > 0. Then R(ζ 2 )f (x) = 2π R R f (y) e iλ(x-y) (λ 2 -ζ 2 ) -1 dy dλ .
Recall the convolution f * e λ (x) = R f (y) e iλ(x-y) dy. This is the Fourier transform

R f * e λ (x) (λ 2 -ζ 2 ) dλ = R-iN f * e λ (x) (λ + ζ)(λ -ζ) dλ -2πi Res λ=-ζ f * e λ (x) (λ + ζ)(λ -ζ) .
The integral on the right is equal to

R-iN f * e λ (x) 2λ(λ + ζ) dλ + R-iN f * e λ (x) 2λ(λ -ζ) dλ .
Furthermore,

R-iN

f * e λ (x) 2λ(λ + ζ) dλ = R+iN f * e λ (x) 2λ(λ + ζ) dλ + 2πi Res λ=0 f * e λ (x) 2λ(λ + ζ) + 2πi Res λ=-ζ f * e λ (x) 2λ(λ + ζ) .
Hence,

R f * e λ (x) (λ 2 -ζ 2 ) dλ = R+iN f * e λ (x) 2λ(λ + ζ) dλ + R-iN f * e λ (x) 2λ(λ -ζ) dλ + πi f * e 0 (x) ζ -πi f * e -ζ (x) ζ + πi f * e -ζ (x) ζ ,
where

f * e 0 (x) = R f (y) dy .
Both integrals on the right-hand side extend to a holomorphic function of ζ for (ζ) > -N , then the following meromorphic continuation of the resolvent of the Laplace operator is

1 2π R(ζ 2 )f (x) = πi f * e 0 (x) ζ + R+iN f * e λ (x) 2λ(λ + ζ) dλ + R-iN f * e λ (x) 2λ(λ -ζ) dλ
So there is just one resonance at ζ = 0, which is a simple pole.

For n > 1, we work in spherical coordinates on λ-coordinate in R n ,

R(z)f (x) = (2π) n +∞ 0 (r 2 -z) -1
S n-1 R n f (y)e irw(x-y) dy dw r n-1 dr .

For a fixed x ∈ R n , let Then F (r) stays rapidly decreasing in r when integrating in w ∈ S n-1 .

F (r) = S n-1 R n f (y)e -
• Suppose n > 2 is odd. Then F is odd. We set ζ = √ z, where the holomorphic branch of the square root function is such that

√ -1 = i. Then (ζ) > 0 and (2π) -n R(ζ 2 )f (x) = +∞ 0 1 r 2 -ζ 2 F (r) r dr = 1 2 +∞ 0 1 r -ζ + 1 r + ζ F (r) dr = 1 2 +∞ 0 F (r) r -ζ dr + 1 2 +∞ 0 F (r) r + ζ dr = 1 2 +∞ 0 F (r) r -ζ dr + 1 2 0 -∞ F (r) r -ζ dr
↑ because F is odd. We cannot do this step when n is even

= 1 2 +∞ -∞ F (r) r -ζ dr .
By Morera's theorem, the above formula extends the function

ζ → R(ζ 2 )f (x)
from the upper half plane to a holomorphic function on C \ R. We want to extend it beyond the upper half plane. We shift the contour of integration in the direction of the negative imaginary axis. By the residue theorem, for R > 0 and N > 0:

R -R F (r) r -ζ dr = -R-iN -R F (r) r -ζ dr + R-iN -R-iN F (r) r -ζ dr + R R-iN F (r) r -ζ dr .
As F is rapidly decreasing, the first and the third integrals on the right-hand side tend to 0 when R goes to infinity. As N is arbitrary, one gets a continuation of

ζ → R(ζ 2 )f (x) to C without singularities.
There are no resonances. The original resolvent z → R(z) has an entire extension to a Riemannian surface which is the branched cover of C associated to the square root.

• Suppose n even. We set r = e t and ζ = log(z), where log(z) is such that its imaginary part is in ]0, π[. We have then In this part we compute the resonances for G = SL(2, R), which will not be considered later. The harmonic analysis is well-known -thanks to the work of Harish-Chandra.

R(e ζ )f (x) = (2π) n +∞ 0 F (e t )
We recall that SL(2, R) is isomorphic to SU(1, 1) and locally isomorphic to SO e (2, 1).

We use a direct method based on the original Harish-Chandra -Plancherel formula for SL(2, R). Will already studied the resonances in this case for SU(1, 1) in [Wil03,

section 5] via a different method. Fix the maximal compact K = SO(2) consisting of elements

k(θ) = cos θ sin θ -sin θ cos θ .
The irreducible unitary representations of K are the characters

χ n : K → C defined by χ n (k(θ)) = e inθ
with θ ∈ R and n ∈ Z. Let A and N be defined as follows

A = t 0 0 t -1 t > 0 , N = 1 y 0 1 y ∈ R .
with Lie algebras

a = x 0 0 -x x ∈ R , n = 0 y 0 0 y ∈ R .
The system of restricted roots is {±α} where α x 0 0 -x = 2x and the half sum of positive roots is ρ = -1 2 α. We have chosen {-α} as positive system for the computations. The centralizer of A in K is M = {±I 2 }. We denote by (π ± iλ , H ± iλ ) the principal series representation Ind G M AN (δ ± ⊗ e iλα ⊗ triv N ), where λ ∈ C and M = {δ + , δ -}, where δ + is the trivial representation and δ -is the non-trivial representation of M . The space H δ ± iλ is the Hilbert space completion of the space of smooth functions f :

G → C such that f (manx) = a -(1+iλ)ρ δ ± (m)f (x) = t 1+iλ δ ± (m)f (x) , where a = t 0 0 t -1 ∈ A, n ∈ N , x ∈ G.
For λ ∈ C, we denote by (π iλ , H iλ ) the representation

H iλ = H + iλ ⊕ H - iλ .

Proposition II.1.2 ([Lan85], VI. §6, Theorem 8, pages 119-124)

If iλ is not an integer, then

H + iλ | K = n∈2Z Ind G K (χ n ) and H - iλ | K = n∈2Z+1 Ind G K (χ n )
are irreducible submodules of H iλ and H iλ is the direct sum of them. If s = 0, then H 0 is the direct sum of three irreducible submodules:

H 0 = n∈2Z Ind G K (χ n ) ⊕ n∈2Z+1 n≥1 Ind G K (χ n ) ⊕ n∈2Z+1 n≤-1 Ind G K (χ n ) .
If m ≥ 2 is an integer and iλ = m -1, then H m-1 contains three irreducible submodules

X m = m≤n n-m∈2Z Ind G K (χ n ) , X -m = -m≥n n-m∈2Z Ind G K (χ n ) , n-m∈2Z+1 Ind G K (χ n ) .

The quotient module (H m-1 divided by the three submodules) is irreducible, finite dimensional of dimension

Trivial representation.

In this thesis, we identify representations if they are infinitesimally equivalent. So we work with (g, K)-modules, when convenient.

The Plancherel theorem of Harish-Chandra for G = SL(2, R) may be stated as follows:

Theorem II.1.1 (Inversion formula for SL(2, R))

For any f ∈ C ∞ c (G), 2πf (g) = ∞ n=1 n Tr X (n+1) + X (-n-1) * f (g) + 1 2 ∞ 0 Tr(π + iλ ) * f (g) λ tanh πλ 2 dλ + 1 2 ∞ 0 Tr(π - iλ ) * f (g) λ coth πλ 2 dλ , (II.4) where (Tr(π) * f ) (g) = G Tr π(x) f (g -1 x) dx
The first term on the right-hand side of (II.4) is the discrete part of the Plancherel formula. We note that this inversion formula is of the same form as (I.15) if we rewrite it in terms of the Fourier transform on G, i.e.

Tr(π + iλ ) * f (g) = Tr π + iλ (g) G π + iλ (x) f (x) dx = Tr π + iλ (g) f (λ) .
A direct computation (see [Lan85,page 195]) shows that in the universal enveloping algebra U (g c )

Ω = 1 0 0 1 2 + 0 1 1 0 2 - 0 1 -1 0 2 and π iλ (Ω) = -(λ 2 + 1) Id .
Let n ∈ Z and χ n ∈ K be fixed. Consider the space

C ∞ c (G, χ n ) of functions f on G such that f (gk) = χ n (k -1 )f (g) , (II.5)
for all k ∈ K and g ∈ G. We denote by φ n,n iλ,± the spherical functions defined by

φ n,n iλ,± (x) = π ± iλ (x)e n , e n , (II.6)
where x ∈ G and e n is as in (II.3). Then the Plancherel formula for SL(2, R) can be

rewritten for f ∈ C ∞ c (G, τ ) as follows 2πf (g) = ∞ l=1 l (φ n,n l * f ) (g) + 1 2 ∞ 0 φ n,n iλ,± * f (g) λ tanh πλ 2 dλ + 1 2 ∞ 0 φ n,n iλ,± * f (g) λ coth πλ 2 dλ . (II.7)
This comes from the identification

f (x) = K χ n (k)f (xk) dk for x ∈ G ,
and the proof is similar to the following proof of (II.10).

As the discrete part of the formula does not give resonances of the Laplace operator, we will omit it in the computations from now.

We consider -Ω acting by the left on C ∞ c (G, χ n ). This action gives a homogeneous differential operator, which is the Laplace operator ∆. This corresponds to the construction of the positive Laplacian ∆ in this thesis in section I.8. One has, for

f ∈ C ∞ c (G, χ n ), ∆(φ n,n iλ,± * f ) = (λ 2 + 1) φ n,n iλ,± * f . (II.8) The resolvent R of ∆ is defined for z ∈ C by R(z) = (∆ -z) -1
We have then, up to a constant, for

f ∈ C ∞ c (G, χ n ) and for all z ∈ C \ [0, +∞[ and g ∈ G (R(z)f ) (g) = ∞ 0 (∆ -z) -1 φ n,n iλ,+ * f (g) λ tanh πλ 2 dλ + ∞ 0 (∆ -z) -1 φ n,n iλ,- * f (g) λ coth πλ 2 dλ = ∞ 0 (λ 2 + 1 -z) -1 φ n,n iλ,+ * f (g) λ tanh πλ 2 dλ + ∞ 0 (λ 2 + 1 -z) -1 φ n,n iλ,- * f (g) λ coth πλ 2 dλ
We change the variable ζ = √ z -1, where the holomorphic branch of the square root function is such that √ -1 = i. Then (ζ) > 0 and

R(ζ 2 + 1)f (g) = ∞ 0 (λ 2 -ζ 2 ) -1 φ n,n iλ,+ * f (g) λ tanh πλ 2 dλ + ∞ 0 (λ 2 -ζ 2 ) -1 φ n,n iλ,- * f (g) λ coth πλ 2 dλ .
Moreover, we have

2λ(λ 2 -ζ 2 ) -1 = 1 λ -ζ + 1 λ + ζ .
Hence, up to a constant,

R(ζ 2 + 1)f (g) = ∞ 0 1 λ -ζ + 1 λ + ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + ∞ 0 1 λ -ζ + 1 λ + ζ φ n,n iλ,- * f (g) coth πλ 2 dλ .
As for R n :

∞ 0 1 λ -ζ + 1 λ + ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ = ∞ 0 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + ∞ 0 1 λ + ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ = ∞ 0 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + 0 -∞ 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ = R 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ ,
Up to a constant,

R(ζ 2 + 1)f (g) = R 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + R 1 λ -ζ φ n,n iλ,- * f (g) coth πλ 2 dλ . (II.9) By Morera's theorem, the function ζ → R(ζ 2 + 1)f (g) is holomorphic in ζ ∈ C \ R.
We want to extend it meromorphically beyond the upper half-plane. For this, we shift the contour of integration in the direction of the negative imaginary axis and apply the residue theorem. The singularities of the meromorphic continuation depend on n being even or odd. For n even, we have:

φ n,n iλ,- * f (g) = G Tr π - iλ (x) f (g -1 x) dx = G K Tr π - iλ (x) f (g -1 xkk -1 ) dk dx = G K Tr π - iλ (xk -1 ) f (g -1 x)χ n (k) dk dx = G Tr π - iλ (x) K π - iλ (k -1 )χ n (k) dk f (g -1 x) dx But K π - iλ (k -1 )χ n (k) dk = 0 because π - iλ | K = j∈2Z+1 χ j .
In the same way, if n is odd,

φ n,n iλ,+ * f (g) = 0 . (II.10)
Hence only one integral in (II.4) is non-zero for f ∈ C ∞ c (G, χ n ), the one which corresponds to the principal series which contains the K-type χ n .

Suppose n is even: The formula (II.9) becomes

R(ζ 2 + 1)f (g) = R 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ (II.11)
For every n ∈ N, the complex number -in is a pole of tanh πλ 2 . Then by the residue theorem, for every N ∈ N + 1/2 and r ∈ R:

r -r 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + r-iN r 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + -r-iN r-iN 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + -r -r-iN 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ = n∈N n<N 1 -in -ζ Tr(π + -n ) * f (g) Res λ=-in tanh πλ 2 .
So we have the following result for r → ∞.

Proposition II.1.3

When n is even :

R(ζ 2 + 1)f (g) = R-iN 1 λ -ζ φ n,n iλ,+ * f (g) tanh πλ 2 dλ + l∈Z * + l<N 1 -il -ζ φ + -l * f (g) Res λ=-il tanh πλ 2 .
When n is odd :

R(ζ 2 + 1)f (g) = R-iN 1 λ -ζ φ n,n iλ,- * f (g) coth πλ 2 dλ + l∈Z * + l<N 1 -il -ζ φ - -l * f (g) Res λ=-il coth πλ 2 .

The above formulas give a meromorphic continuation of the resolvent of the positive

Laplace operator on the half-plane above R -iN with simple poles at ζ = -il where • l ∈ Z * + , l < N and l odd for n even

• l ∈ Z * + , l < N and l even for n odd We make N → +∞. Let M = {(z, ζ) ∈ C 2 | z = ζ 2 + 1}.

II.1.3 Rank-one case (excluding SL(2, R))

In this section, we prove Theorem 1. We recall that the resonances of the positive Laplace operator ∆ are defined as the poles of the meromorphic continuation of its resolvent (∆-z) -1 considered as an operator defined on C ∞ c (G, τ ). We know thanks to Lemma I.9.2, that the spherical functions ϕ σ,λ τ are eigenfunctions of ∆ with eigenvalues M (σ, λ) := λ, λ + ρ, ρ -µ σ + ρ m , µ σ + ρ m . The Plancherel theorem (I.8) (or more explicitly in (I.15)) gives us a decomposition of L 2 (G, τ ) into a continuous and, possibly, a discrete part:

L 2 (G, τ ) = L 2 cont (G, τ ) ⊕ L 2 dicr (G, τ
) By a slight abuse of notation, we identify R(z) with its restriction to the set

L 2 cont (G, τ )∩ C ∞ c (G, τ
). The reader will notice that the discrete part we are omitting brings no additional resonances.

Lemma II.1.1 Let z ∈ C \ [ ρ, ρ , +∞[. The function R(z) can be written R(z) = 1 d τ σ∈ M (τ ) R σ (ζ σ ) (II.12) for all f ∈ C ∞ c (G, τ ) where R σ (ζ σ )f (x) = 1 |α| R 1 λ|α| -ζ σ ϕ σ,λα τ * f (x) p σ (λα) λ dλ (II.13)
and ζ σ is defined by

ζ σ := z -ρ, ρ + µ σ + ρ M , µ σ + ρ M ) . (II.14) Proof. Let f ∈ C ∞ c (G, τ ).
Due to the inversion of the vector-valued Helgason-Fourier transform (I.15) and the formula (I.23) for the convolution product between ϕ σ,λ τ

and f ∈ L 2 cont (G, τ ) ∩ C ∞ c (G, τ ), we have f (x) = 1 d τ σ∈ M (τ ) a * ϕ σ,λ τ * f (x) p σ (λ) dλ
Because of Lemma I.9.3, we know that ϕ σ,λ τ * f is a rapidly decreasing smooth function. So by Lemma 2.2,

R(z)f = 1 d τ σ∈ M (τ ) a * (∆ -z) -1 ϕ σ,λ τ * f (x) p σ (λ) dλ = 1 d τ σ∈ M (τ ) a * (M (σ, λ) -z) -1 ϕ σ,λ τ * f (x) p σ (λ) dλ .

II.1. Computation of the resonances

Computing R(z) is then equivalent to compute for each σ ∈ M (τ )

R σ (z) := a * (M (σ, λ) -z) -1 ϕ σ,λ τ * f (x) p σ (λ) dλ
As the values of µ σ , ρ and ρ M are constant we can introduce the variable ζ σ defined in (II.22). Finally, changing the variable with the isomorphism (I.4) between a * C and C:

R σ (ζ σ )f (x) := R (λ 2 |α| 2 -ζ 2 σ ) -1 ϕ σ,λα τ * f (x) p σ (λα) dλ
Notice that we are using the same symbol λ for the variable in a * and the corresponding value λ α ∈ R. Since the function ϕ σ,λα τ is even in λ the same computation as in [HP09, Lemma 2.4] yields the formula (II.13):

R σ (ζ σ )f (x) = R 1 2λ|α| 1 λ|α| -ζ σ + 1 λ|α| + ζ σ ϕ σ,λα τ * f (x) p σ (λα) dλ = 1 2|α| R 1 λ|α| -ζ σ + 1 λ|α| + ζ σ ϕ σ,λα τ * f (x) p σ (λα) λ dλ = 1 |α| +∞ 0 1 λ|α| -ζ σ + 1 λ|α| + ζ σ ϕ σ,λα τ * f (x) p σ (λα) λ dλ = 1 |α| R 1 λ|α| -ζ σ ϕ σ,λα τ * f (x) p σ (λα) λ dλ Morera's theorem ensures that the function R σ ( • )f (x) is holomorphic in ζ σ ∈ C\R.
We want to extend meromorphically R σ ( • )f (x) from one half-plane to the other. To fix notation, we will consider R σ ( • )f (x) as a function defined on the upper half plane (ζ) > 0. We will find the meromorphic continuation of this function to C by shifting the contour of integration in the direction of the negative imaginary axis and by applying the residue theorem. The poles of the Plancherel density give then poles of the meromorphic continuation. That is why, if the Plancherel density has no poles, so does the meromorphic continuation, and the Laplacian has no resonances (as for G = Spin(2n + 1, 1)). The formula of Plancherel density is given in Appendix A (equations (A.1), (A.2), (A.4) and (A.5)) for each of the rank-one groups G. The following proposition unifies the case-by-case formulas found in the literature. The proof is a direct computation remplacing the values in each case thanks to the table in the following Remark.

Proposition II.1.4

Let G be of real rank-one. Set

m α := 1 2 (m α/2 + m α -1) .
Suppose s even:

The formula above goes in tanh(πλ), which has first order poles at λ ∈ i Z + 1 2 however the singular points of this function are the imaginary numbers of the form i Z + 1 2 . Since s is even, the B j + ρ -j are in Z + 1 2 . Hence the zeros of the polynomial part of pσ(λα) λ are the element

{±i(B j + ρ -j) | j = 1, . . . , m α } . (II.19)
Thus the Plancherel density has simples poles in the complement of this set in i(Z + 1 2 ).

Suppose s odd:

The formula above goes in coth(πλ), with simple poles at λ ∈ iZ. Since s is odd, the B j + ρ -j are in Z. Thus the poles of the Plancherel density are simple and located in the complement of the set (II.19) in iZ.

-i(B 1 + ρ -1) • -i(B 2 + ρ -2) • • -i(B 3 + ρ -3) • -i(B m α -2 + ρ -m α + 2) • -i(B m α -1 + ρ -m α + 1) • -i(B m α + ρ -m α ) • • • • poles Figure II.1: Example of poles with B 2 = B 3 + 1 for SO(2n, 1) Let B max = max(|B 1 + ρ -1|, |B m α + ρ -m α |) .
(II.20)

All the values of the form -i(B max + k), with k ∈ N, are poles of the Plancherel density. Let us view R σ as the D (X)-valued holomorphic function on (ζ σ ) > 0 defined in (II.13). We want to determine the meromorphic continuation of this function through the real axis. For this we shift the contour of integration in the direction of the negative imaginary axis as in Figure II.2 below. Due to Corollary I.9.1, the convolution product is rapidly decreasing in λ. Then we just need to upper bound the expression (ζ σ -λ|α|) -1 pσ(λα) λ by a polynomial in |λ|, to make the two integrals along the vertical segments between -R and -R-i(ρ 1 +N +1/2) and between R and R-i(ρ 1 +N +1/2) tend to 0 when R goes near infinity. For | (λ)| near to infinity and (ζ σ ) > 0, we have

|(ζ σ -λ|α|) -1 | < (ζ σ ) -1 and pσ(λα) λ < (1 + |λ|) deg(pα)+1
, where p α is the polynomial part of p σ . So, the shift is allowed for all N ∈ N.

-i(B 1 + ρ -1) • -i(B 2 + ρ -2) • • -i(B 3 + ρ -3) • -i(B m α -2 + ρ -m α + 2) • -i(B m α -1 + ρ -m α + 1) • -i(B m α + ρ -m α )) • • • • R • -R • -i(ρ 1 + N + 1 4 )
• poles

(R σ (ζ σ )f ) (x) = 1 |α| R-i(N+1/4) 1 λ|α| -ζ σ ϕ σ,λα τ * f (x) p σ (λα) λ dλ + 2iπ |α| k∈Nσ λ k >-i(N +1/4) 1 λ k |α| -ζ σ ϕ σ,λ k α τ * f (x) Res λ=λ k p σ (λα) λ (II.21)
The right-hand side of this formula yields a meromorphic continuation of the resolvent of the Laplace operator on (ζ) > -(N + 1/4). The singular values of the Plancherel density induce then poles of the meromorphic continuation of the resolvent of the Laplace operator. If we resume the notation in the expression (II.22

): z = -ζ 2 σ - ρ, ρ + µ σ + ρ M , µ σ + ρ M .
This proves the theorem.

Theorem II.1.2. Let G be a connected non-compact semisimple Lie group with finite center and with Iwasawa decomposition G = KAN , where K is a fixed maximal compact subgroup of G. Suppose dim A = 1. Let M denote the centralizer of A in K. Let (τ, V τ ) be an irreducible unitary representation of K, and let E τ be the homogeneous vector bundle over G associated with τ .

ζ σ := -z -ρ, ρ + µ σ + ρ M , µ σ + ρ M ) (II.22) with z ∈ C such that (ζ σ ) > -(N + 1/4).
Here √

• denotes the single-valued branch of the square root function determined on C \ [0, +∞[ by the condition √ -1 = -i. The resonances of the Laplace operator acting on the sections of E τ appear in families parametrized by the elements of M (τ ). Let

S σ = (z, ζ) ∈ C 2 | ζ 2 := -z -ρ, ρ + µ σ + ρ M , µ σ + ρ M ) .

Then the resolvent extends meromorphically from S

+ σ = {(z, ζ) ∈ S | (ζ) > 0} to S σ .

The (simple) poles of this extension are the pairs

(z σ,k , ζ σ,k ) = (B max +k) 2 |α| 2 -ρ 2 α |α| 2 + µ σ +ρ M , µ σ +ρ M , -i(B max +k) 2 |α| 2 (II.23)
where k ∈ N σ (defined below the figure II.2), µ σ is the highest weight of the representation σ, B max is defined in (II.20) and ρ M is half sum of roots of M .

II.2 Residue representations

We consider C ∞ (G, τ ) as a G-module by left-translations. One can see that for each σ ∈ M (τ ) and for each k ∈ N σ , the residues at λ k in the meromorphic continuation

(II.21) span a G-invariant subspace of C ∞ (G, τ ) if f ∈ C ∞ c (G, τ )
. This is exactly the image of the G-intertwining map :

R σ k : C ∞ c (G, τ ) -→ C ∞ (G, τ ) f -→ ϕ σ,λ k τ * f . (II.24)
We denote this space by

E σ k := {ϕ σ,λ k τ * f | f ∈ C ∞ c (G, τ )} . (II.25)
We want to identify these representations in terms of their Langlands parameters, decide which of them are unitarizable and compute their wave front sets. The idea is to decompose R σ k as follows: Recall the notation (π σ λ k , H σ λ k α ) for the principal series representation corresponding to σ and iλ k α. For all l = 1, . . . , m(σ, τ | M ), we denote by P l the projection of H σ λ k α on its l-th K-isotypic component, which we identify with (τ, V τ ). We get the following decomposition of the residue operator R σ k as a composition of two G-intertwining maps:

R σ k : C ∞ c (G, τ ) → H σ λ k α → C ∞ (G, τ ) f → T l (f ) → l P l π σ λ k α ( • -1 ) T l (f ) (II.26)
where

T l is the map from C ∞ c (G, τ ) to H σ λ k α defined by T l (f ) = G π σ λ k (g) P * l f (g) dg .
(II.27)

Here * denotes the Hermitian adjoint. Hence, P * l maps V τ into the principal series and is K-equivariant.

Lemma II.2.1 T l is an intertwining operator between the left regular representation of G on C ∞

c (G, τ ) and the principal series representation (π σ λ k , H σ λ k α ). Moreover, for each l the range of the map T l is the closed subspace of H σ λ k α spanned by the left translates of P * l V τ . We will denote this space by

π σ λ k (G)P * l V τ .
Proof. First of all, by definition,

T l C ∞ c (G, τ ) is contained in π σ λ k (G)P * l V τ . Fix an element g 0 of G and a vector v 0 in V τ . Let δ g 0 ∈ C c (G) * be the (scalar) Dirac delta at g 0 and set δ g 0 ,v 0 = δ g 0 v 0 . Consider the operator on C ∞ c (G, τ ) defined by as the distribution f → G K τ (k)δ g 0 ,v 0 (gk)dk f (g)dg. Since P *
l is a K-intertwining operator between τ and π λ k , calculations show that

T l K τ (k)δ g 0 ,v 0 ( • k) dk = π σ λ k (g 0 )P * l v 0 .
Since the Dirac delta can be approximated by smooth compactly supported functions, each element of π σ λ k (G) P * l V τ can be written as limit of elements of T l C ∞ c (G, τ ) . This proves the lemma.

Remark

The map from

H σ λ k α to C ∞ c (G, τ ) defined by φ → P l π σ λ k ( • -1 )φ is a G-intertwining operator between (π σ λ k , H σ λ k α
) and the left regular representation on C ∞ c (G, τ ). It is known as the Poisson transform (see [START_REF] Olbrich | Die poisson-transformation für homogene vektorbündel[END_REF][START_REF] Yang | Poisson transforms on vector bundles[END_REF]).

The idea on how to identify E σ k is to compute the range of the map T l , for each l, using Lemma II.2.1. Knowing the composition series of H σ λ k α , we can identify this range in this principal series and then project back with the Poisson transform, the second part of the map in (II.26). The main issue is that, even in rank-one, the structure of the principal series representations is very complicated in general. See [START_REF] David | Representations of rank one Lie groups[END_REF]. In this thesis we will consider only the case when σ is the trivial representation of M . In this case the composition series is more transparent and the result have a pleasant uniform form. For example, τ occurs in π σ λ k with multiplicity one. So, from now on, we fix an irreducible representation τ ∈ K which contains the trivial representation of M . We study only the residue representations which arise from σ = triv M and we will denote them E k := E triv M k . Similarly, we set R k := R triv k and write T and P instead of T l and P l respectively. The map in (II.26) becomes:

R k : C ∞ c (G, τ ) → H triv λ k α → C ∞ (G, τ ) f → T (f ) → P π triv λ k α ( • -1 ) T (f ) (II.28)
where P is the projection onto the τ -isotypic component. The structure of the spherical principal series representations (π λ , H λ ) := (π triv λ , H triv λ ) of our groups G has been studied by different authors (see e.g. [HT93, JW77, Joh76, Mol99]) for every G we are studying. Our main reference will be the paper of Howe and Tan [START_REF] Howe | Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations[END_REF] which provides an explicit description of the subquotients of this principal series representation. We will treat the residue representations case by case. In each case, we compute the

Langlands parameters of E k , k ∈ N σ .
Method to compute explicitly the Langlands parameters: First of all we have to compute the minimal K-type of E k in the sense of Vogan [START_REF] Vogan | Representations of real reductive Lie groups[END_REF]5.4.18.]. It can be done by minimizing Vogan's norm on the set of K-types in E k . Using the branching rules in [START_REF] Baldoni | Branching theorems for semisimple lie groups of real rank one[END_REF] we find the representation δ. In fact, from the highest weight of the minimal K-type τ min we know what are the representations in its restriction to M . Then we induce back from these representations and we check if τ min is the minimal K-type in these induced representations. For G = Sp(n, 1) this work is already done in Proposition 3.1 in [START_REF] Baldoni | The unitary dual of Sp(n, 1), n ≥ 2[END_REF]. Sometimes such δ does not exist. This implies that π is a discrete series representation. [Vog81, Theorem 6.6.15] ensures us that the two representations H triv λ k and H δ ν are conjugate under K. This means that the infinitesimal characters of π triv λ k α and π δ ν are the same up to one element of the Weyl group of the pair (g C , h C ). They can be computed depending on ν. Recall that π σ λ has infinitesimal character µ σ + ρ m + λ where µ σ is the highest weight of σ and ρ m is half sum of positive roots orthogonal to the restricted root α. Up to a sign change, there is a unique possible value of ν. The residue representation E k can then be identified with the unique Langlands subquotient containing the same minimal K-type τ min . To summarize:

1. Find a minimal K-type τ min of π.

2. Find δ such that τ min is minimal in Ind K M (δ). If this doesn't work then π is a discrete series representation.

3. Find ν such that the infinitesimal character of H triv λ k is the same as the infinitesimal character of H δ ν . This means that µ triv + ρ m + λ k is conjugate to µ σ + ρ m + ν by an element of the complex Weyl group.

II.2.1 Example of R

Recall that in the case of the R the Laplace operator (or the double derivative in this case) has only one resonance at λ = 0 (see section II.1.1). In this case, the residue representation takes place is the space

x → R f (y) dy | f ∈ C ∞ c (R) ,
which is the set of constant functions. R acts trivially on this set. So the residue representation here is the trivial representation of R.

II.2.2 Example of SL(2, R)

Recall (see section II.1.2) that here

φ n,n l (x) = π l (x)e n , e n , (II.29)
where x ∈ G, e n is as in (II.3) and (π l , H l ) is defined in Proposition II.1.2. Then we obtain the following residue representations

When n is even :

{φ -l * f | l is odd, f ∈ C ∞ c (G, χ n )} .
When n is odd :

{φ -l * f | l is even, f ∈ C ∞ c (G, χ n )} .
So l -1 and n have the same parity. Then π -l (G)P * V χn is one of the following irreducible components of π -l (see also Proposition II.1.2)

X -l-1 = p≤-l-1 p+l-1∈2Z Ind G K (χ p ) , X l+1 = p≥l+1 p+l-1∈2Z Ind G K (χ p ) , -l-1<p<l+1 p+l-1∈2Z
Ind G K (χ p ) .

(II.30) In particular, π -l (G)P * V χn is the component where χ n belongs:

1. π -l (G)P * V χn = X -l-1 if n ≤ -l -1, 2. π -l (G)P * V χn = X l+1 if n ≥ l + 1 3. π -l (G)P * V χn is the final dimensional constituent above if -l -1 < n < l + 1
For a fixed number n, we have |n| 2 -1 infinite dimensional representations which are isomorphic to the holomorphic discrete series X -l-1 or X l+1 (as in [Wil03, Theorem 5.1]). The other representations are finite dimensional.

II.2.3 Case of SO(2n, 1)

Let G = SO(2n, 1). The K-types of the spherical principal series representations are parametrized by m ∈ N and known as the space of spherical harmonics on R 2n of homogeneous degree m, denoted by H m (R 2n ). Their highest weight of the form m 1 with respect to the fundamental weights described in section A.1. A representation τ containing the trivial representations of M is of this form. From now on, let τ act on the harmonic polynomials on R 2n of fixed homogeneous degree N :

V τ H N (R 2n ) .
As σ is trivial the poles of the Plancherel density p triv M are the

λ k = -i(ρ α + k) = -i(n + k -1/2)
with k ∈ N. The composition series of H λ k α described in [START_REF] Howe | Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations[END_REF] is the following:

H λ k α 1 k m=0 P * m H m (R 2n ) ⊕ 2 m>k P * m H m (R 2n ) (II.31)
where P m is the projection of

H λ k α onto the K-isotypic component isomorphic to H m (R 2n
). The action of G cannot send a K-type from the second summand to the first one. The second map in (II.26) (the Poisson transform) is an intertwining operator. So its kernel is either 0 or the entire representation. Choosing a nonzero h ∈ H N (R 2n ) nonzero, the function P τ N π λ k α ( • -1 )j -2n-k+1 h has value h at e G . So the kernel of the Poisson transform is not the entire space, thus it is 0. Consequently E k is 2 in (II.31).

Langlands parameters of E

0 triv K • • • • • • • • • k + 1 • • • • • N • • • • • • • Image of T triv K • • • • • • • • • k + 1 • • • • • • • • • • • • Image of T N 0 Kernel of
Suppose that N ≤ k: Here the image of T is the entire representation H λ k α because the action of g can cross the barrier at k + 1. The kernel of the second intertwining map in (II.28) is 2 in (II.31) by the same argument as before. It follows that E k is the subquotient isomorphic to 1 in (II.31).

The unitarity is given in [START_REF] Howe | Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations[END_REF]Diagram 3.15]. To identify the representations we have determined, we compute their Langlands parameters. They are collected in Table II

.1. Case Minimal K-type δ values of ν N > k + 1 H k+1 (R 2n ) H k+1 (R 2n-1 ) ± n -3 2 α N ≤ k triv K triv M (ρ α + k)α
Table II.1: Langlands parameters of E k when G = SO(2n, 1)

The entries of Table II.1 are computed as follows. Let µ l be the highest weight of H l (R 2n ). A minimal K-type minimizes the Vogan norm of the highest weight in E k :

µ l V = µ l + 2ρ K , µ l + 2ρ K
where ρ K is half sum of positive roots in S k C + (see Appendix A.1 for the definition). One can check that µ l V is minimal when l is minimal. This yields the first column of the table. , for 1 µ δ (k + 1) , for 2 .

To find ν one has to compare the infinitesimal character of Ind G M AN (triv ⊗ e (ρα+k)α ⊗ triv) and Ind G M AN (δ ⊗ e ν ⊗ triv) .

They have to agree up to the action of the Weyl group of (g C , k C ). Theorem 2, 1., follows from these computations.

Theorem II.2.1. Suppose that the representation τ contains the trivial representation of M . The residue representations E k are then irreducible.

• If N ≥ k+1, then E k has Langlands parameters M A, H k+1 (R 2n-1 ), n -3 2 α with (k + 1, 0, . . . , 0) as a lowest K-type's highest weight. Here H k+1 (R 2n-1 ) are harmonics of degree k + 1 on R 2n-1 . This representation is unitary.

• If N < k + 1, then E k has Langlands parameter M A, triv, (ρ α + k) with the trivial representation as a lowest K-type. It is finite-dimensional. Also, it is non-unitary if k = 0.

II.2.4 Case of SU(n, 1)

Let now G = SU(n, 1). The structure of the spherical principal series is described for U(n, 1) in [START_REF] Howe | Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations[END_REF] but Molchanov find the same result for SU(n, 1) in [START_REF] Molchanov | Representations of pseudo-unitary groups associated with a cone[END_REF]. Choose a basis {z 1 , . . . , z n , z n+1 } of C n+1 as a vector space and {z 1 , . . . , z n , z n+1 , z 1 , . . . , z n , z n+1 } the corresponding basis of R 2n+2 . The K-types of the spherical principal series representations are the spaces

H m 1 ,m 2 (C n ) ⊗ H l (C)
where H m 1 ,m 2 (C n ) are the spherical harmonics on C 2n of homogeneous degree m 1 and m 2 in the variables z 1 , . . . , z n+1 and z 1 , . . . , z n , z n+1 respectively. Moreover, the space H l (C) is defined for all integer l by:

H l (C) = H l,0 (C) if l ≥ 0 H 0,-l (C) if l ≤ 0
Their highest weights are of the form m 1 1 -m 2 n -l n+1 with respect to the fundamental weights described in section A.2. From now on, let

V τ H M 1 ,M 2 (C n ) ⊗ H L (C) .
(II.32) so the highest weight of τ is

µ τ = M 1 1 -M 2 n -L n+1 (II.33)
for fixed nonnegative integers M 1 , M 2 , a fixed integer L. From now on we call this representation τ M 1 ,M 2 ,L . As σ is trivial, the poles of the Plancherel density are the

λ k = ±i( n 2 + k)
where k is a non-negative integer. Hence

H λ k α l∈Z m 1 ,m 2 ≥0 m 1 -m 2 +l=0 P * H m 1 ,m 2 (C n ) ⊗ H l (C) (II.34)
where P := P m 1 ,m 2 ,l is the projection of H λ k α onto the K-isotypic component isomorphic to H m 1 ,m 2 (C n ) ⊗ H l (C). One can see that the conditions in the sum in (II.34) imply that the pair (m 1 + m 2 , l) determines the triple (m 1 , m 2 , l) and since m 1 and m 2 are nonnegative, -l ≤ m 1 + m 2 ≤ l for all l ∈ Z. The reader should recall Lemma II.2.1 and (II.28).

• τ M 1 ,M 2 ,L Kernel of the Poisson transform Image of T 1 m l • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -2k -2 2k + 2 2 m l • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -2k -2 2k + 2 • 3 m l • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -2k -2 2k + 2 • 4 m l • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • -2k -2 2k
Case 1 :

M 1 + M 2 ≥ 2k + 2 and |L| ≤ -2k -2 + M 1 + M 2
Since the action of G on the K-types cannot cross the barriers from the side out, the image of T is the entire East-component

m 1 ,m 2 ≥0, m 1 +m 2 ≥2k+2 l∈Z, |l|≤-2k-2+m 1 +m 2 m 1 -m 2 +l=0 P * H m 1 ,m 2 (C n ) ⊗ H l (C) .
The Poisson transform is an intertwining operator. Hence its kernel, contained in an irreducible representation, is either 0 or the entire representation. Choosing a nonzero h ∈ H N (R 2n ), the function

P τ M 1 ,M 2 ,L π λ k α ( • -1 )P * (h) has value h at e G .
Thus E k is equivalent as a representation to the image T .

Case 2 : L ≥ | -2k -1 + M 1 + M 2 | + 1
From the North-constituent, the action of G can cross the barrier l = -2k -2+m but not the barrier l = 2k + 2 -m. Thus the image of T is composed of two North and East-components:

m 1 ,m 2 ≥0 l∈Z, l>2k+2-m 1 -m 2 m 1 -m 2 +l=0 P * H m 1 ,m 2 (C n ) ⊗ H l (C) .
Following the proof in the previous case, one can conclude that the image of the Poisson transform is nonzero and that the North-constituent, where τ M 1 ,M 2 ,L is, is not in the kernel of this map. Because of the barrier l = -2k -2 + m the action of G cannot bring a K-type from the East-constituent into the Northconstituent. Thus the East-constituent is the kernel of the Poisson transform and E k is equivalent to the subquotient of the North-East constituents modulo the East-constituent.

Case 3 : L ≤ | -2k -1 + M 1 + M 2 | -1
This case is completely symmetric to the previous one. The figure explains the result.

Case 4 : where -(k + 1) ≤ a ≤ k + 1. One can prove that µ = a 1 -a 3 is always the highest weight of a K-type in Ind K M (δ) with a smaller Vogan norm than τ min . So we get then a discrete series representation. The Blattner parameter of the discrete series is the highest weight µ τ min of its minimal K-type τ min . Its Harish-Chandra parameter is

M 1 + M 2 ∈ [0, 2k + 2[ and L < |2k + 2 -M 1 + M 2 | τ M 1 ,M 2 ,
Λ k = µ τ min + 2ρ k -ρ g = (k + 1) 1 -k 2 -3 .
To find ν, one has to compare the infinitesimal characters of Ind G M AN (triv ⊗ e (ρα+k)α ⊗ triv) and Ind G M AN (δ ⊗ e ν ⊗ triv) .

They have to coincide up to the action of the Weyl group of (g C , k C ). For Case 1 , one gets respectively the infinitesimal characters

n 2 + k ( 1 -n+1 ) + 1 2 n j=2 (n -2i + 2) i and ν α ( 1 -n+1 ) + n 2 + k 2 + 1 2 n-1 j=3 (n -2i + 2) i + - n 2 -k n
The (complex) Weyl group permutes the ± i 's, so ν = ± n 2 -1 α. The theorem follows from similar computations, in other cases.

Theorem II.2.2. We suppose that τ is a K-type containing the trivial representation of M . τ has a highest weight of the form (M 1 , 0, . . . , 0, -M 2 , -L), where M 1 and M 2 are positive integers such that M 1 ≥ M 2 ≥ 0, L ∈ Z and M 1 + M 2 + L is even. The residue representations E k , defined in (II.25), are irreducible.

1 If M 1 + M 2 ≥ 2k + 2 and |L| ≤ -2k -2 + M 1 + M 2 ,
the residue representation is unitary.

• if n > 2, this representation has Langlands parameter M A, δ, ± n 2 -1 α where the highest weight of δ is (k + 1), 0, . . . , 0, -(k + 1), 0 .

• if n = 2, this representation is the discrete series with Blattner parameter ((k + 1), (k + 1), 0).

2 If L ≥ | -2k -1 + M 1 + M 2 | + 1, the residue R k is the representation with Langlands parameter M A, δ, ± k 2 + n 2 -1 2 α
, where the highest weight of δ is 0, 0, . . . , 0, -(k + 1), (k + 1)/2 . This representation is non-unitary.

3 If L ≤ | -2k -1 + M 1 + M 2 | -1, the residue R k is the representation with Langlands parameter M A, δ, ± k 2 + n 2 -1 2 α
, where the highest weight of δ is (k + 1), 0, . . . , 0, (k + 1)/2 . This representation is non-unitary.

4 If M 1 + M 2 ∈ [0, 2k + 2[ and L < |2k + 2 -M 1 + M 2 |,

Remark

Since the multiplicity of τ in H δ ν is 1 (see for instance [START_REF] Koornwinder | A note on the multiplicity free reduction of certain orthogonal and unitary groups[END_REF]) for these first two cases, the residue representation can be identified as the unique irreducible subquotient of H δ ν containing τ .

II.2.5 Case of Sp(n, 1)

Let G = Sp(n, 1). We recall that K = Sp(1) × Sp(n). Up to equivalence there a unique representation of Sp(1) = SU(2) of dimension j + 1 for all nonnegative integers j. Denote this representation by θ j , acting on the space V j 1 . Denote by V m 1 ,m 2 p the irreducible representation of Sp(p) with highest weight (m 1 , m 2 , 0, . . . , 0) where m 1 ≥ m 2 ≥ 0 (see [Ž73, Theorem 6 page 327]).

As σ is trivial, the poles are the points

λ k = ±i(n + 1 2 + k)
with k ∈ N. The spherical principal series representation decomposes over K as follows:

H P * V m 1 ,m 2 p ⊗ V l 1 (II.35)
where

P := P m 1 ,m 2 ,l is the projector in H λ k α on the K-isotopic component isomorphic to V m 1 ,m 2 p ⊗ V l
1 and the sum is over the following set :

m 1 ≥ m 2 ≥ 0, l ≥ 0, m 1 + m 2 ≥ l, m 1 + m 2 + l ∈ 2Z .
(II.36)

These conditions imply that a fixed pair (m, l) = (m 1 + m 2 , l) represents a fiber of K-

types V m 1 ,m 2 p ⊗ V l 1 , one for each value of m 1 -m 2 . The highest weight of V m 1 ,m 2 p ⊗ V l
1 can be computed as m 1 1 + m 2 2 + |l| n+1 with respect to the sets of roots in (A.3).

Let τ be V a 1 ,a 2 p ⊗ V an+1

1

. From now on we call this representation τ a 1 ,a 2 ,a n+1 .

Langlands parameters of E k . To understand the composition series of the representation E k , we have now to know how p acts on its K-types. This action is given in [HT93, Lemma 5.3]. The diagram 5.18 in that paper gives us the different cases.

One can prove that the action of g can bring one K-type in each fiber to every other in the same fiber. Then we can follow the method used for G = SO(2n, 1) or SU(n, 1) putting the K-types of H λ k α in the same two-dimensional space corresponding to the points with coordinates (m 1 + m 2 , l). Notice that here each point in Figure II.5 corresponds then to a fiber of K-types. We refer to [HT93, Lemma 5.4] for more details. Figure II.5 illustrates the computation of the residue representation E k . Two barriers cross the set of K-types. These separate the space H λ k α in three constituents. The figure shows the three cases where the K-type τ a 1 ,a 2 ,a n+1 is in each constituent. The arguments are the same as in the complex case. Naming the constituents North, West ant East according to their position, one gets the following equivalences:

E k                  East-constituent if a n+1 ≤ -2k -4 + a 1 + a 2 North-East-constituents East-constituent if a n+1 ≥ | -2k -2 + a 1 + a 2 | H λ k α North-East-constituents if a n+1 ≥ 2k + 2 -a 1 -a 2 (II.37) • τ a 1 ,a 2 ,a n+1 Kernel of the Poisson transform Image of T m l • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2k + 4 2k + 2 m l • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2k + 4 2k + 2 • m l • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2k + 4 2k + 2 • Figure II.5: K-types in H λ k α and decomposition series for k = 1
The three representations E k we have found above are irreducible subquotients of H λ k α . Their unitarity is given in [HT93, Diagram 3.15]. We find the following results using Proposition 3.1 in [START_REF] Baldoni | The unitary dual of Sp(n, 1), n ≥ 2[END_REF]:

Case Minimal K-type δ values of ν 1 (k + 2) 1 + (k + 2) 2 (k + 2)e 3 + (k + 2)e 4 ± n -3 2 α 2 (k + 1) 1 + (k + 1) n+1 (k + 1)e 3 + k+1 2 (e 1 -e 2 ) if k ≤ 2n -4 ∅ if k > 2n -4 ± k 2 + n α if k ≤ 2n -4 3 triv K triv M (ρ α + k)α
Table II.3: Langlands parameters of E k when G = Sp(n, 1) We indicate how to compute the entries of this table. In case 2 , for k > 2n -4, one cannot find a representation δ following the conditions of the Langlands parameters. We conclude that E k is in the discrete series in these cases. Their Blattner parameter is the highest weight of the minimal K-type, which is (k + 1) 1 + (k + 1) n+1 . The Harish-Chandra parameter of the discrete series is

(k + n) 1 + (k + 3 -n) n+1 + n j=2 2(n -i + 1) i .
Theorem II.2.3. We suppose that τ is a K-type containing the trivial representation of M . τ has a highest weight of the form (t 1 , t 2 , 0, . . . , 0, t n+1 ), where t 1 , t 2 and t n+1 are non-negative integers such that t 1 ≥ t 2 , t n+1 = t 1 -t 2 and t 1 + t 2 + t n+1 is even. The residue representations E k , defined in (II.25), are irreducible. 1 If t n+1 ≤ t 1 + t 2 -2k -4 , the residue E k is the representation with Langlands parameter M A, δ, ± n -3 2 α with lowest K-type's highest weight (k + 2, k + 2, 0, . . . , 0) and where the highest weight of δ is (k + 2, k + 2, 0, . . . , 0). This representation is non-unitary.

2 If t n+1 ≥ |t 1 + t 2 -2k -2| the residue representation is unitary. • If k ≤ 2n-4, the residue E k is the representation with Langlands parameter M A, δ, ± n -k 2 α with lowest K-type (k + 1, 0, . . . , 0, k + 1)
, where the highest weight of δ is (k + 1, 0, . . . , 0, k+1 2 ). • If k ≥ 2n -3, the residue E k is the discrete series representation with Blattner parameter µ k = (k + 1, 0, . . . , 0, k + 1).

3 If t 1 + t 2 < 2k + 2 -t 1 -t 2 , the residue E k is the representation with Langlands parameter M A, triv, (ρ α + k)α . It is finite-dimensional and non unitary (if k = 0).

Remark

One can verify that Theorem 1 in [START_REF] Parthasarathy | Classification of discrete series by minimal K-type. Represent[END_REF] gives the same conclusion as ours: there are discrete series representations in case 2 for k > 2n -4. In fact, to verify the hypotheses of the theorem, we take the set of non-compact positive roots being

∆ + n = { 1 ± n+1 , n+1 ± i for i > 1} . We get 2ρ n = 2 1 + (2n -2) n+1 , such that, λ = µ k -2ρ n = (k -1) 1 + (k + 3 -2n) n+1 ,
which is dominant if and only if (k + 3 -2n) ≥ 0. This gives the same condition over k.

II.2.6 Case of F 4

The paper of Johnson [START_REF] Johnson | Composition series and intertwining operators for the spherical principal series[END_REF] gives the results we need in this case. Let G be the exceptional Lie group F 4 . We recall that here K = Spin(9) and M = Spin(7). As σ is trivial, the poles of the meromorphically extended resolvent are the points

λ k = ±i( 11 2 + k)
with k ∈ N.

The K-types of H λ k α are the V p,q with p ≥ q ≥ 0 and p + q ∈ 2Z (see [Joh76, Theorem 3.1]), with highest weight

µ p,q = p 2 1 + q 2 2 + q 2 3 + q 2 4
with respect to the sets of roots in Appendix (A.3). Let τ be V a,b , for a ≥ b ≥ 0 and a + b ∈ 2Z. In the following, we call this representation τ a,b . We can follow the method of the cases G = SO(2n, 1) or SU(n, 1), putting the Ktypes of H λ k α in the same two-dimensional space corresponding to the points with coordinates (p, q). Figure II.6 illustrates the computations of E k . There are again two barriers. They separate the space H λ k α in three constituents. The figure describes the three cases where the K-type τ p,q is in different constituents. The arguments are the same as in the complex case. Naming the constituent North, West ant East according to their position, one gets the following equivalences:

E k                  East-constituent if a n+1 ≤ -2k -4 + a 1 + a 2 North-East-constituents East-constituent if a n+1 ≥ | -2k -2 + a 1 + a 2 | H λ k α North-East-constituents if a n+1 ≥ 2k + 2 -a 1 -a 2 (II.38) • τ a,b Kernel of the Poisson transform Image of T p q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2k + 8 2k + 2 p q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2k + 8 2k + 2 • p q • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2k + 8 2k + 2 • Figure II.6: K-types in H λ k α and decomposition series for k = 0
The three representations E k we have found above are irreducible subquotients of H λ k α . We find the following results using [BS79, Theorem 3.4]:

III Wave front set of the residue representations

In this section, we continue to assume that (τ, V τ ) contains the trivial representation of M . The residue representations E k are those listed in Theorem 2 and defined in section II.2. The purpose of this part is to compute the wave front set WF(E k ) of these representations. For this, we need some results on the Gelfand-Kirillov dimension [START_REF] Vogan | Gelfand-Kirillov dimension for Harish-Chandra modules[END_REF] and on the nilpotent orbits in semisimple Lie algebras [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF].

III.1 The wave front set of a representation

First of all, wave front sets were introduced in P.D.E.'s. They allow to quantify the smoothness of a distribution. They were adapted to representation theory by Roger Howe in [START_REF] Howe | Wave front sets of representations of Lie groups[END_REF]. Proposition 2.4 in [START_REF] Howe | Wave front sets of representations of Lie groups[END_REF] tells us that the wave front set of E k is equal to a closed union of nilpotent orbits of g. To know which occur, we use the Gelfand-Kirillov dimension of E k , computed by [Vog78, Theorem 1.2] in terms of Ktypes. This dimension is half of the dimension of the wave front set seen as a union of nilpotent orbits (see [START_REF] Vogan | Gelfand-Kirillov dimension for Harish-Chandra modules[END_REF][START_REF] Rossmann | Picard-Lefschetz theory and characters of a semisimple Lie group[END_REF][START_REF] Barbasch | The local structure of characters[END_REF]). Combining this information with the list of nilpotent orbits in semisimple Lie algebras and their dimensions in [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF], we identify the wave front set in Theorem 2.

In general, a compactly supported distribution f is smooth if and only if its Fourier transform f is rapidly decreasing and of exponential type. This is the Paley-Wiener theorem, well known for functions on open subsets of R n .

Lemma III.1.1 (Paley-Wiener Theorem for R n )

Let f be compactly supported distribution on R n . The following assertions are equivalent:

1 f ∈ C ∞ c (R n ) ,
2 for all N ∈ N * and ξ ∈ R n , we have

| f (ξ)| ≤ C N (1 + |ξ|) -N (III.1)
where C N is a constant depending on N . This means that if f is not smooth, there exist some directions in R n , where f is not rapidly decreasing. Let us then introduce the set Σ(f ) of all η ∈ R n \ {0} having no conic neighborhood V on which (III.1) is true. Σ(f ) is then a closed cone in R n \ {0} and as soon as f is not smooth, this cone is not empty. It describes only the directions of high frequencies causing singularities of f . For each

x in an open subset U of R n , we can define Σ x (f ) = ψ Σ(ψf ) ,
where the intersection is over all ψ ∈ C ∞ c (U ) such that ψ(x) = 0. The wave front set of a function f on an open subset U of R n can then be defined as follows

WF(f ) := {(x, η) ∈ U × (R n \ {0}) | η ∈ Σ x (f )} . This is a closed cone of U × (R n \ {0}).
1 The question of the dimension is answered by [CM93, Corollary 6.1.4.]. It gives the dimensions for the complex nilpotent orbits in g C . The dimensions of the real nilpotent orbits are the same (but there is not the same number of orbits).

2 Theorems 9.3.3, 9.3.4 and 9.3.5 in [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF] allow us to decide how many orbits there are in each case. The exceptional case is treated in section 9.6.

III.3 Case of SO(2n, 1)

In this section, we prove the results about wave front sets stated in Theorem 2 for G = SO(2n, 1). First we compute the Gelfand-Kirillov dimension of the Harish-Chandra module of E k defined in section II.2. This case contains the case of SL(2, R).

Lemma III.3.1

The Gelfand-Kirillov dimension of the residue representation

E k is    0 if N ≤ k 2n -1 if N > k
Proof. First of all, if N ≤ k, the residue representation is finite-dimensional, so the Gelfand-Kirillov dimension is 0. It follows from definition of the Gelfand-Kirillov dimension. For the case N > k, the residue representation is infinite-dimensional. One can compute the eigenvalue of each K-type τ m in H λ k α for the Casimir operator Ω k of k (using for example [Hal15, Proposition 10.6]):

τ m (Ω k ) = µ τm + ρ k , µ τm + ρ k -ρ k , ρ k Id = (m + n -1) 2 -(n -1) 2 Id ,
where ρ k = n j=1 (n -j) j is half sum of positive roots in (k, h| k ). We recall that here τ m is the space of harmonic polynomial of homogeneous degree m. One can compute its dimension (see [Ž73] for example)

d m = m + 2n -1 2n -1 - m + 2n -3 2n -1 .
According to Vogan's Theorem (Proposition III.1.2), we compute the sum N E k (t) of the dimensions d m of τ m until its eigenvalue exceeds a fixed real number t 2 . We have

(m + n -1) 2 -(n -1) 2 ≤ t 2 ⇐⇒ t→∞ m + n -1 ≤ ∞ t
So we will sum the dimensions up to N t = t -n + 1. This sum of dimensions (depending on t) is then written as follows:

N E k (t) = Nt m=k+1 d m = N t + 2n -1 2n -1 + N t + 2n -2 2n -1 - k + 2n -2 2n -1 - k + 2n -1 2n -1 , (III.8)
because the sum is telescopic. If t goes to infinity, we have :

N E k (t) ∞ ∼ N t + 2n -1 2n -1 ∞ ∼ (N t + 2n -1)! N t ! ∞ ∼ t 2n-1 (III.9)
We denote above by ∞ ∼ the fact that the higher degree in t is the same on the right and the left-hand side. Hence the Gelfand-Kirillov dimension is equal to 2n -1.

Lemma III.3.2 Let g = m ⊕ a ⊕ g α ⊕ g -α (III.10)
be the restricted root space decomposition of g. Then there are 2 nilpotent orbits in g, namely the zero orbit and the orbit generated by any non-zero element of g α .

Proof. The non-zero elements of g α are conjugate by the elements of M A. Moreover, g -α = θ(g α ) = Ad(k θ )(g α ) for a suitable element k θ ∈ K. Then all non zero nilpotent elements in the restricted root spaces above are conjugate. This can also be found using Theorem 9.3.4 in [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF]. For each real nilpotent orbit in so(2n, 1), it is written that there is one signed (alternating '+' and '-' on each row) Young diagram with 2n '+' and one '-' such that the rows of even length occurs with even multiplicity. Here even length is not possible because there is just one '-'. There are only two possible Young diagrams for so(2n, We are now able to prove the results about the wave front set of E k in Theorem 2 for G = SO(2n, 1).

Theorem III.3.1. The wave front set of the residue representation E k is either the zero orbit or the nilpotent orbit generated by g α , respectively when k ≥ N or k < N . We recall that N is the nonnegative integer which defines the highest weight (N, 0, . . . , 0) of τ .

Wave front set of E k . We have just two cases. When N ≤ k, E k is finite-dimensional and the Gelfand-Kirillov dimension is 0. The wave front set is then the zero orbit. If N > k, as E k is infinite-dimensional, only the nilpotent orbit generated by g α can correspond.

This can be checked using the formula for the dimension of nilpotent orbits in g C , the complexification of g ([CM93, Corollary 6.1.4]). In fact if N > k, the Gelfand-Kirillov dimension is 2n -1 because of Lemma III.3.1. The dimension of the wave front set is then 4n -2. Because of the corollary cited above, we have:

4n -2 = dim WF(E k ) = (2n + 1) 2 - 1 2 s 2 i - 1

III.4 Case of SU(n, 1)

In this section, we prove the results about wave front sets in Theorem 2 for G = SU(n, 1). First of all, we compute the Gelfand-Kirillov dimension of the Harish-Chandra module of E k defined in Section II.2. Recall that there are 4 possibilities for the residue representation (see II.2.4). We label each case as in

E k is            2n -1 in case 1 n in case 2 n in case 3 0 in case 4 .
Proof. In case 4 , the representation E k is finite-dimensional. The Gelfand-Kirillov dimension is then 0. For the three other cases, we need some computations.

First of all, one can compute the eigenvalue of each K-type τ m,l in H λ k α for the Casimir operator Ω k of k (using for example [Hal15, Proposition 10.6]). Recall from II.2.4 that

τ m,l H m-l 2 , m+l 2 (C n ) × H l (C) (III.12) with highest weight µ m,l = m-l 2 1 -m+l 2 n + l n+1 .
Then one finds:

τ m,l (Ω k ) = µ τ m,l + ρ k , µ τ m,l + ρ k -ρ k , ρ k Id = 1 2 (m + n -1) 2 -(n -1) 2 + 3l 2 Id , (III.13)
where

ρ k = n j=1 (n-2i+1 2
i is half sum of positive roots in (k, h| k ). The dimension of the space of τ m,l follows for example from the Weyl dimension formula [Kna02, Theorem 5.84]:

d m,l = α∈Σ + µ m,l + ρ k , α α∈Σ + ρ k , α = m-l 2 + n -2 n -1 m+l 2 + n -2 n -1 (n -1)(m + n -1) m-l 2 m+l 2
As we want to sum the dimensions, we will make appear a telescopic sum over m, as in the previous case. One can verify that the dimension can be rewrite as

d m,l = m-l 2 + n -1 n -1 m+l 2 + n -1 n -1 - m-l 2 + n -2 n -1 m+l 2 + n -2 n -1
The sum over m is now telescopic each 2 integers. We recall that for a fixed l the values of m change by 2 integers because l + m has to be even. The inequality given by the eigenvalue of the Casimir operator (III.13) limits the area of points of K-types by an ellipsis. The goal is to know the behaviour in t of the sum over the K-types inside the ellipsis when this t-ellipsis is going to infinity.

t 2 = 1 2 (m + n -1) 2 -(n -1) 2 + 3l 2 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •
First one can see, that for a fixed l, the sum between two values m min and m max of m is telescopic: (III.15) So on each (red) l-line, the sum of the dimensions of each representation (point) is just the difference between values depending on the first (blue) and the last (green) representation. these colored points represent respectively the representations τ m min ,l and τ mmax ,l . We compute the sum N E k (t) of the dimensions d m of τ m until the eigenvalue (III.13) exceeds a fixed real number t 2 . We have for the eigenvalues

(m + n -1) 2 -(n -1) 2 + 3l 2 ≤ t 2
So the values of l and m are bounded by a certain l max and m max respectively, depending on t. In term of power of t, the sum can be then written :

N E k (t) ∞ ∼ lmax (t) l=-lmax (t)
δ(m max (l, t), l, n)

In fact, we can forget the term δ(m min , l, n -1) because it is a constant in t so it will bring strictly lower powers of t than δ(m max , l, n) in the sum. We recall that we denote above by ∞ ∼ the fact that the higher degree in t is the same on the right and the left-hand side. We will now compute case by case.

Case 1 : The sum N E k (t) is then over the ellipsis (figure above), whose 'radius' is going to infinity in t 2 . The number l max (t) is roughly equal to the ordinate of the intersection between the ellipsis and the line "limiting" the representation (the 45°line). Computing the intersection, we have that l max (t) is asymptotic at infinity to t 2 . As m max (l, t) 2 is roughly on the ellipsis, we also have that m max (l, t) is asymptotic at infinity to t 2 -3l 2 on the line with ordinate l. We get then

N E k (t) ∞ ∼ lmax (t) l=-lmax (t) δ(m max (l, t), l, n) ∞ ∼ |l|≤t/2
n-1 j=1 (mmax (l,t)+n-1-j) 2 4

l 2

4

(n -1)!

And replacing m max :

N E k (t) ∞ ∼ 0≤l≤t/2 (t 2 -4l 2 ) n-1 ∞ ∼ t 2n-1
The Gelfand-Kirillov dimension follows from Proposition III.1.2. The cases 2 and 3 are symmetric.

Case 2 : The sum N E k (t) is then over the ellipsis , with 'radius' going to infinity in t 2 , intersected with the area between 2 parallel lines "limiting" the representation (the 45°-lines). The number l max (t) is roughly equal to the ordinate of the intersection between the ellipsis and the line "limiting" the representation (the 45°-line). Computing the intersection, we have that l max (t) is asymptotic at infinity to t 2 . As m max (l, t) is roughly on the ellipsis or on the second "limiting" line, we also have that m max (l, t) 2 is asymptotic at infinity to t 2 -3l 2 or to l on the line with ordinate l. So m max (l, t) is of the same power of t at infinity. The difference with the preceding case is that the binomial coefficient in m -l in δ(m max (l, t), l, n) is always a constant on the second "limiting" line. We get then

N E k (t) ∞ ∼ lmax (t) l=-lmax (t) δ(m max (l, t), l, n) ∞ ∼ |l|≤t/2 t n-1 ∞ ∼ t n .
The Gelfand-Kirillov dimension follows from Proposition III.1.2.

The following lemma gives us the different nilpotent orbits in the nilradical of g.

Lemma III.4.2

There are four nilpotent orbits in SU(n, 1): Here the 0 in the center of the matrix is the (n -1) × (n -1) zero matrix.

1 the trivial orbit,
Proof. Theorem 9.3.3 in [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF] ensures us that (real) nilpotent orbits in su(n, 1) are parametrized by signed (alternating '+' and '-' on each row) Young diagram with 2n '+' and one '-'. One finds that there are four possible Young diagrams corresponding to the nilpotent orbits:

+

The first one corresponds to the zero orbit. The second one is the only one which is nilpotent of degree 2. It corresponds to the orbit generated by any element of g α/2 . In fact, the proof of the fact that g α/2 and g -α/2 are in the same orbit is exactly the same as the one in Lemma III. For the partition [2, 1, 1, . . . , 1], the dimension is

(n + 1) 2 - n i=1 s 2 i = (n + 1) 2 -n 2 -1 = 2n .
This completes the dimensions of the nilpotent orbits.

We are now able to prove the results about the wave front set of E k in Theorem 2 for SU(n, 1).

Theorem III.4.1. The representation τ has highest weight of the form (M 1 , 0, . . . , 0, -M 2 , -L), where M 1 and M 2 are positive integers such that M 1 ≥ M 2 ≥ 0, L ∈ Z and M 1 + M 2 + L is even. Wave front set of E k . The two lemmas above conclude the cases 1 and 4 . In fact, there is one possibility of dimension of nilpotent orbit corresponding to each Gelfand-Kirillov dimension. Now we have to know which of the cases 2 and 3 correspond to the nilpotent orbit generated by n 1 or to the one generated by n 2 . They have the same dimension, so we need to have more information on wave front sets. We will use the projection over K to figure it out. In fact, Proposition III.1.3 ensures us that

1 If M 1 + M 2 ≥
WF(E k | K ) = Ad * (K) -AC(supp E k ) (III.16)
where we recall that AC the asymptotic cone and supp ρ is the set of highest weights of E k . Proposition 2.5 gives us:

WF(E k | K ) = q(WF E k ) (III.17)
where q is the projection from g * onto k * . We write E 2 k or E 3 k if E k is respectively in case 2 or in case 3 (see figure III.1). We have:

-AC(supp E 2 k ) -µ 1,1 = n -n+1 and -AC(supp

E 3 k ) µ 1,-1 = -1 + n+1 Moreover, q • B : n 1 → (2n + 2)(-1 + n+1 ) n 2 → (2n + 2)( 1 -n+1 )
This gives the result, as K sends by the coadjoint action n -n+1 to 1 -n+1 .

III.5 Case of Sp(n, 1)

In this section, we prove the results on wave front sets, in Theorem 2 for G = Sp(n, 1). First of all, we compute the Gelfand-Kirillov dimension of the Harish-Chandra module of E k defined in section II.2. Recall that there are 3 possibilities of residue representation (see II.2.5). We label each case as in Figure III.2. The proofs are very similar to the complex case. So we will be less specific. The inequality given by the eigenvalue of the Casimir operator III.13 limits the area of points of K-types by an ellipsis. The goal is to know the behaviour in t of the sum over the K-types delimited by this ellipsis when this ellipsis is going to infinity. The method to compute the N E k (t) out of [Vog78, Theorem 1.2] is exactly the same as the SU(n, 1) case and left to the reader.

A The Plancherel densities

Recall that to determine the resolvent of the Laplacian, we use the inversion formula (I.15) for vector-valued Helgason-Fourier transform. The resonances arise from the singularities of the Plancherel measure. To find them, we need an explicit formula for the Plancherel density p σ . Even if such an explicit formula is not known for an arbitrary group G, in the rank-one case, several authors have computed it ([Oka65], [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF]; see also [War72, Epilogue, pp. 414 ff.]). Our reference in the following is Miatello's article [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF]. The formula depends on the group G and on the highest weight of σ. According to this formula, p σ is the product of two factors. The first one is a polynomial function in λ, denoted q σ . The second factor is either a hyperbolic tangent or a hyperbolic cotangent (or 1 if G = Spin(2n + 1, 1)). We denote it by φ σ . The goal of this section is to explain how one can compute p σ with σ ∈ M (τ ), where τ ∈ K is arbitrarily fixed. For this, we are going to use the branching rules given by Baldoni Silva in [START_REF] Baldoni | Branching theorems for semisimple lie groups of real rank one[END_REF]. Remark 1.3 in [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF] ensures us that the Plancherel formula of one of the four groups listed in the table in the introduction gives us the Plancherel formula for every group of real rank one. As we do not care about the constants in our computations in this thesis, p σ is given up to a constant.

A.1 Case of Spin(n, 1)

Here G/K is the real hyperbolic space. Recall that K is Spin(n) and M is Spin(n -1). In this case, the parity of n plays a role in the Plancherel measure. In fact, when n is odd, φ σ (λ) = 1. So it is non-singular and there are no resonances. In the following we therefore disregard the case of Spin(2n + 1, 1) and suppose that G = Spin(2n, 1). Let us recall some Lie algebraic structure. Our reference is [BS79, Section 3]. The Lie algebra of G is g = so(2n, 1). Its complexification is g C = so(2n + 1, C). We have also Lie(K) C = k C = so(2n, C) and Lie(M ) C = m C = so(2n -1, C).

We choose the following Cartan subalgebra of g C is:

h C = H ∈ so(2n + 1, C) | H = diag 0 ih 1 -ih 1 0 , 0 ih 2 -ih 2 0 , . . . , 0 ih n -ih n 0 , 0
Let { j } j=1,...,n be the elementary weights defined by j (H) = h j . We denote S, S + and S 0 respectively the set of roots, positive roots and simple roots of (g C , h C ). We index the sets of roots by the Lie algebra, to which they correspond. Notice that, unlike [BS79, Theorem 3.4], we keep the same notation for the j 's and their projections on m C . As in [BS79, Lemma 3.2], the fixed (τ, V τ ) ∈ K has highest weight of the form:

S = ± i ± j | i = j ∪ ± k | k ∈ {1, . . . , n} ; S 0 = α i = i -i+1 | i ∈ {1, . . . , n-1} ∪ α n = n S k C = ± i ± j | i = j ; S m C = ± i ± j | i = j ∪ ± k | k ∈ {1
µ τ = n j=1 a j j
where a 1 ≥ . . . ≥ a n-1 ≥ | a n | ≥ 0, a i -a j ∈ Z and 2a j ∈ Z for all i, j = 1, . . . , n.

Let σ ∈ M (τ ). According to [BS79, Theorem 3.4], the highest weight µ σ of a representation σ ∈ M (τ ) has the form:

µ σ = n-1 i=1 b j j
where for all i, j = 1, . . . The Cartan subalgebra used in [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF] is neither t nor h because the maximal abelian subalgebra of p is not the same a here. Define the map Ψ :

α 1 → α 2 α 2 → α 1 α 2 + 2α 3 + α 4 → α 3 α 3 → -(α 2 + 2α 3 + α 4 )
This map Ψ sends α 4 on -1 , so a onto the maximal abelian subspace used in [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF].

The Cartan subalgebra Ψ(h) is that used in that paper. Applying Ψ we get

µ σ•Ψ = b 1 1 + b 2 2 + b 3 3
and 1 becomes the real root, i.e. 1 | Ψ(a) = -α • Ψ where α is the longest restricted root. Now we can apply the Plancherel formula from [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF] and get We recall that λ α is the complex number associated to λ ∈ a * C by (I.4).

p σ (λ) = tanh(πλ α ) , if b i ∈ Z coth(πλ α ) , if b i ∈ Z + 1 2 λ α λ 2 α + ( b 3 + 1 2 ) 2 λ 2 α + ( b 2 + 3 2 ) 2

II. 1

 1 Example of poles with B 2 = B 3 + 1 for SO(2n, 1) . . . . . . . . . . . . II.2 Shift of contour and residue theorem for SO(2n, 1) . . . . . . . . . . . . II.3 K-types in H λ k α and composition series for k = 7 . . . . . . . . . . . . II.4 K-types in H λ k α and composition series for k = 1 . . . . . . . . . . . . II.5 K-types in H λ k α and decomposition series for k = 1 . . . . . . . . . . . II.6 K-types in H λ k α and decomposition series for k = 0 . . . . . . . . . . . III.1 Cases for E k -SU(n, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . III.2 Cases for E k -Sp(n, 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . III.3 Cases for E k -F 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  ) with z ∈ C such that (ζ σ ) > -(N + 1/4). Here √ • denotes the single-valued branch of the square root function determined on C \ [0, +∞[ by the condition √ -1 = -i. The resonances of the Laplace operator acting on the sections of E τ appear in families parametrized by the elements of M (τ ). Let

  then E k is the representation with Langlands parameter M A, triv, (ρ α + k)α . It is finite-dimensional and non unitary (if k = 0). 4. If G = F 4 , then τ has a highest weight of the form (a/2, b/2, b/2, b/2), where a and b are positive integers such that a ≥ b and a -b is even.

  for every H ∈ a} contains nonzero elements. The dimension of g α is called the multiplicity of the root α and is denoted by m α . In other words, Σ = {α ∈ a * | g α = 0} (I.1) If α ∈ Σ then -α ∈ Σ. We can choose an order: Let Σ + be a fixed set of positive restricted roots and let ρ := 1 2 α∈Σ + m α α be the half sum of the positive roots counted with their multiplicities. Set n = α∈Σ + g α and N the connected Lie subgroup of G having n for Lie algebra. The Iwasawa decomposition of the Lie algebra can be then written g = k ⊕ a ⊕ n. According to the Iwasawa decomposition at the group level G = KAN , every element x in G can be uniquely written as x = k(x)e H(x) n(x) (I.2) where k(x) ∈ K, H(x) ∈ a and n(x) = n ∈ N . In the following, we set a λ := exp(λ(log a)) for a ∈ A and λ ∈ a * C . (I.3)

  The spherical function ϕ σ,λ τ is defined as the End(V τ )-valued function on G given byϕ σ,λ τ (x) := ϕ π σ λ τ (x) := d τ K τ (k)ψ σ,λ τ (xk -1 )dk , (I.18) where ψ σ,λ τ (x) = Tr P τ π σ λ (x)P τ . (I.19)

  positive. Its spectrum is [0, +∞[. Then the resolvent of ∆, R(z) = (∆ -z) -1 is a bounded linear operator which depends holomorphicaly on z ∈ C\[0, +∞[. The restriction of R(z) to a dense subspace of L 2 (R n ), for example the space of compactly supported smooth functions C ∞ c (R n ), has a meromorphic extension accross [0, +∞[:

  of a smooth compactly supported function. It extends to a Paley-Wiener function on the complex plane. The function f * e λ (x) (λ 2 -ζ 2 ) has two singularities: at λ = ζ and λ = -ζ. For any N > (ζ) only one of them, λ = -ζ, is between the two horizontal lines, R and R -iN . Therefore the residue theorem implies that for any N > (ζ),

  irwy dy = f (rw) is rapidly decreasing in r by the Paley-Wiener theorem e irwx dw r n-2 .

  e 2t e 2t -e 2ζ dt . The same method as before allows us to extend ζ → R(e ζ )f (x) as an entire function on C. There are also no resonances. The original resolvent z → R(z) has an entire extension to a Riemannian surface which is the branched cover of C associated to the logarithm. Proposition II.1.1 There are no resonances of the Laplace operator ∆ = -n j=1 ∂ 2 ∂x 2 j acting on smooth compactly support functions on R n , when n > 1. For n = 1, there is one resonance at 0. II.1.2 Example of G = SL(2, R)

  Then the resolvent extends meromorphically from M + = {(z, ζ) ∈ M | (ζ) > 0} to M and with simple poles at (-l + 1, -il), l ∈ Z * + . These poles are the resonances of the Laplace operator.

Figure II. 2 :

 2 Figure II.2: Shift of contour and residue theorem for SO(2n, 1)

  k . In Figure II.3 below, each bullet corresponds to one K-type of the representation H λ k α , the abscissa of the bullet being the coefficient appearing in the highest weight of the K-type. The figure describes the two cases N ≥ k and N < k. The barrier at k + 1 and the arrows mean that the action of g cannot send a K-type which is on the right of k + 1 to a K-type which on the left of k + 1.

  Figure II.3: K-types in H λ k α and composition series for k = 7Suppose that N ≥ k + 1: The image of T is the space spanned by the action of g on P * m H N (R 2n ) . Since that action cannot cross the barrier at k + 1, the image of T is the infinite-dimensional irreducible representation

  Finally, to find δ, one can use [BS79, Theorem 3.4]. The M -types in H l (R 2n ) have highest weight µ δ (a) = a 1 where a ∈ [[0, k+1]]. The only one which has the same minimal K-type is µ δ (0)

+ 2 •Figure

 2 Figure II.4: K-types in H λ k α and composition series for k = 1

  the residue R k is the representation with Langlands parameter M A, triv, (ρ α + k)α . It is finitedimensional and non unitary (if k = 0).

  See [H 03][8.1 p.852].

  +', which respectively correspond respectively to the 0 orbit and the orbit generated by g α .

FigureFigure III. 1 :

 1 Figure III.1: Cases for E k -SU(n, 1)

  l = δ(m, l, n) -δ(m, l, n -1) = δ(m, l, n) -δ(m -2, l, n) . (III.14)

  , l, n) -δ(m -2, l, n) = δ(m max , l, n) -δ(m min , l, n -1) .

2

  the one generated by g α/2 , of dimension 4n -2, 3 the one generated by n 1 = i

  3.2.The two last ones are nilpotent of degree 1. Computations show that n 1 and n 2 (defined in Lemma III.3.2) cannot be conjugated by an element of K. In particular, K cannot change the last number on the last line and the last column. In fact, let k = B 0 0 e iθ , where B ∈ SU (2n), θ ∈ R and det(B)e iθ = 1. We have:Ad(k)n 1 =where b 11 is the first coefficient in B. The conjugation by A is a multiplication by a positive factor and the conjugation by N doesn't affect n 1 or n 2 . So n 1 and n 2 are each representative of one nilpotent orbit of degree 1. The dimensions are given by Corollary 6.1.4 in[START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF]. Recall that the partition of the complex nilpotent orbit coming from a real nilpotent orbit is given by the boxes of the corresponding Young diagram. One can compute, for the first one [3, 1, 1, . . . , 1]. The numbers s i = |{j | d j ≥ i}| are (n -1), 1, 1, 0, . . ., 0. So the dimension of the orbit

Figure III. 2 :1 -m-l 2 n

 22 Figure III.2: Cases for E k -Sp(n, 1)

  , . . . , n -1} S 0 k C = i -i+1 | i ∈ {1, . . . , n -1} ; S 0 m C = i -i+1 | i ∈ {1, . . . , n -2} ∪ n-1

  , n -1 we have a j -b i ∈ Z and a 1 ≥ b 1 ≥ a 2 ≥ . . . ≥ a n-1 ≥ b n-1 ≥ | a n | ≥ 0. Furthermore, m(σ, τ | M ) = 1 for every σ ∈ M (τ ).For σ ∈ M (τ ), the Plancherel density is given in[START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF]:p σ (λ) = tanh(πλ α ) , if b j ∈ Z coth(πλ α ) , if b j ∈ 1 2 + Z λ α n-1 j=1 λ 2 α + (ρ + b j -j) 2 (A.1)We recall that λ α is the complex number associated to λ ∈ a * C by (I.4).A.4. Case of F 4whered 1 ≥ b 1 . . . ≥ b 3 ≥ |d 4 |, 2c i ∈Z and a i -c j ∈ Z for all i, j = 1, . . . , 4. Applying φ, one getsµ σ = b 1 α 2 + (b 1 + b 2 )α 1 + 1 2 (b 1 + b 2 + b 3 )(α 2 + 2α 3 + α 4 )

Corollary 4.0.1

  alors E k est la représentation avec le paramètre de Langlands M A, triv, (ρ α + k)α . Il est alors τ a un plus haut poids de la forme (t 1 , t 2 , 0, . . . , 0, t n+1 ), où t 1 , t 2 et t n+1 sont des entiers positifs tels que t 1 ≥ t 2 , t n+1 ≤ t 1 +t 2 et t 1 +t 2 +t n+1 est pair.Pour un k ∈ N fixe, il existe une correspondance biunivoque entre les sous-quotients irréductibles de H λ k α et les orbites nilpotentes (réelles) de g sous l'action adjointe.

	de dimension finie et non unitaire (si k = 0). Cette correspondance envoie chaque sous-quotient dans l'orbite dont la fermeture est
	l'ensemble du front d'onde de ce sous-quotient. Comme nous l'avons montré, E k est équivalent à l'un de ces sous-quotients. Son ensemble de front d'onde est alors la 3. Si G = Sp(n, 1), • Si t n+1 ≤ t 1 + t 2 -2k -4, alors E k est la représentation avec le paramètre fermeture d'une orbite nilpotente dans g.
	de Langlands M A, δ, ± n -3 2 α avec τ comme K-type minimal, où le
	plus haut poids de δ est (k + 2, k + 2, 0, . . . , 0). Cette représentation n'est
	pas unitaire. Son front d'onde est l'orbite nilpotente générée par g α/2 .
	• Si t n+1 ≥ |t 1 + t 2 -2k -2| la représentation résiduelle est unitaire. Son
	ensemble de front d'onde est l'orbite nilpotente générée par g α .
	-Si k ≤ 2n -4, alors E k est la représentation avec le paramètre de Lang-
	lands
	M A, δ, ± n -k 2 α avec le plus petit K-type (k + 1, 0, . . . , 0, k + 1),
	où le plus haut poids de δ est (k + 1, 0, . . . , 0, k+1 2 ).
	-Si k ≥ 2n -3, alors E k est la représentation en série discrète avec le
	paramètre Blattner µ k = (k + 1, 0, . . . , 0, k + 1).
	• Si t 1 +t 2 < 2k +2-t 1 -t 2 , alors E k est la représentation avec le paramètre de
	Langlands M A, triv, (ρ α + k)α . Il est de dimension finie et non unitaire
	(si k = 0).
	4. Si G = F 4 , alors τ a un plus haut poids de la forme (a/2, b/2, b/2, b/2), où a et
	b sont entiers positifs tels que a ≥ b et ab sont pairs.
	• Si b ≤ a -2k -8, alors E k est la représentation avec le paramètre de Lang-
	lands
	M A, δ, 1 2 (k + 1) α , où le plus élevé de δ est k+4 4 (3, 1, 1, 1). Cette représen-
	tation n'est pas unitaire. Son front d'onde est l'orbite nilpotente générée par
	g α/2 .
	• Si b > a -2k -8 et b ≥ 2k + 2 -a, alors E k est la représentation avec
	le paramètre de Langlands M A, δ, 1 2 (k + 10) α , où le plus élevé de δ est
	k+1 4 (3, 1, 1, 1). Cette représentation est unitaire. Son front d'onde est l'orbite
	nilpotente générée par g α .
	• Si b < 2k + 2, alors E k est la représentation avec le paramètre de Langlands
	M A, triv, (ρ α + k)α . Il est de dimension finie et non unitaire (si k = 0).
	Une belle conséquence de ces résultats au cas par cas est la suivante.

  Ce -m+2 ⊕ Ce -m+4 ⊕ ... ⊕ Ce m-2 .

	where e n is the function on G defined by
	e n	u 0 0 u	y x 0 1	cos θ sin θ -sin θ cos θ	= y 1+iλ e inθ .	(II.3)
	2. Principal series H + iλ and H -iλ , λ ∈ R \ {0};
	3. Principal series H + 0 ;
	4. Complementary series H + s , 0 < s < 1;

m -1. The (g, K)-modules H m-1 and H -m+1 are dual to each other.

If m ≥ 2 is an integer and iλ = -m + 1, then H m-1 contains the finite dimensional submodule Here u, y and x are real such that the indicated product is in G. Thus we have the highest weight modules, lowest weight modules, finite dimensional modules and modules with unbounded K-types on both side.

Here is the complete list of the irreducible unitarizable (g, K)-modules 1. Lowest weight module X m with lowest weight m ≥ 1 and the highest weight module X m with highest weight m ≤ -1

  L is in the finite-dimensional West-constituent. From there, the action of G can send a K-type onto any other K-type in H λ k α . Thus the image of T is the entire spherical principal series representation. As in the previous cases, one can prove that the image of the Poisson transform is nonzero and that the Westconstituent, where τ M 1 ,M 2 ,L is, is not in the kernel of this map. Because of the two barriers, the action of G cannot bring a K-type from the other constituents into the West-constituent. This means that the kernel of the Poisson transform

  = |{j | d j ≥ i}| and r i = |{j | d j = i}| in a partition [d 1 , . . . , d k ] of 2n + 1.Thus, only one complex nilpotent orbit has this dimension: the one matching to the partition [3, 1, 1, . . . , 1] of 2n + 1. This corresponds indeed to the Young diagram of the (real) nilpotent orbit generated by g α .

	2 odd	r i	(III.11)
	where s i

  2k + 2 and |L| ≤ -2k -2 + M 1 + M 2 , then the wave front set of E k is the nilpotent orbit generated by g α/2 .2 If L ≥ | -2k -1 + M 1 + M 2 | + 1,then the wave front set of E k is the nilpotent orbit generated by the element n 2 of g α (see Lemma III.4.2 for the definition). 3 If L ≤ | -2k -1 + M 1 + M 2 | -1, then the wave front set of E k is the nilpotent orbit generated n 1 of g α (see Lemma III.4.2 for the definition). 4 If M 1 + M 2 ∈ [0, 2k + 2[ and L < |2k + 2 -M 1 + M 2 |, then the wave front set of E k is the zero orbit.

Remerciements :

Abstract

We study the resonances of the Laplacian acting on the compactly supported sections of a homogeneous vector bundle over a Riemannian symmetric space of the noncompact type. The symmetric space is assumed to have rank-one but the irreducible representation τ of the maximal compact K defining the vector bundle is arbitrary. We determine the resonances. Under the additional assumption that τ occurs in the spherical principal series, we determine the resonance representations. They are all irreducible. We find their Langlands parameters, their wave front sets and determine which of them are unitarizable.

Then the Plancherel density is given by the following formula:

where

and

Here the b j 's are the coefficients of the highest weight of σ. We refer the reader to Appendix A to see the computations of this formula.

If one wants to use this formula, one has to find which representations σ ∈ M appear in τ , identifying their highest weight thanks to Appendix A, and then compute the Plancherel measure for each of these σ.

Remark

To make the computations in each of the four cases, one needs the following table:

is the quotient

The four representations E k we have found above are irreducible subquotients of H λ k α . Their unitarity is given in [START_REF] Howe | Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations[END_REF]Diagram 3.15]. We list their Langlands parameter in Table II.2.

To find δ, one can use [START_REF] Baldoni | Branching theorems for semisimple lie groups of real rank one[END_REF]Theorem 4.4]. We show the reasoning in case 1 . The minimal K-type τ min has highest weight

Suppose n = 2: The difference with the case n > 2 is that there is no integer between 1 and n. Here

, where the highest of δ is k+4 4 (3, 1, 1, 1). This representation is non-unitary.

, where the highest of δ is k+1 4 (3, 1, 1, 1). This representation is unitary.

• If b < 2k + 2, then E k is the representation with Langlands parameter M A, triv, (ρ α + k)α . It is finite-dimensional and non unitary (if k = 0).

For functions on a Lie group, the wave front set is a closed cone of the cotangent bundle T * (G) = G × g * . Howe introduced this notion to representation theory [How81, page 118], as follows. Let (π, V π ) be a representation of G. Let T the space of all trace class operators on the Hilbert space V π . For T ∈ T , we define the continuous function

Then Tr π is a bounded norm-decreasing function. Its image is usually called the space of (continuous) matrix coefficients of π. We also define the distribution

(Wave front set of a representation)

We define the wave front set WF(π) of a representation (π, V π ) of G as the closure in T * (G) of the union of WF Tr π (T ) as T varies over T .

As it is invariant under left and right translation of G, we identify WF(π) with its projection on g * . It turns out to be a closed conical Ad * G-invariant subset.

The wave front set of a representation is an interesting object, in the sense that this is not always g * (it could be). Among others, it gives information about the differential and the "size" of the representation. We recall some important results of [START_REF] Howe | Wave front sets of representations of Lie groups[END_REF], we are using in this thesis:

Proposition III.1.1 (1.5 in [How81])

Let H be a closed subgroup of G with Lie algebra h. Let us denote by q the restriction map from g * to h * . We have then the inclusion:

Using this proposition for H = K, the maximal compact of G, and for an irreducible representation π of G, this proposition becomes more precise:

5(a) in [How81])

Let us denote by q the restriction map from g * to k * . Then

Let V be a vector space. We recall that the asymptotic cone AC of a nonempty set A ⊂ V is defined by the two equivalent conditions:

2 For all u ∈ U , if any cone containing a neighbourhood of u intersects A in an unbounded set, then u ∈ AC(A).

III.2. Nilpotent elements of g

If σ ∈ K, we denote the set of highest weights of σ by supp σ.

Proposition III.1.3 (2.3 in [How81])

For a unitary representation σ of K, we have

Using these propositions, we are going to decide what is the wave front set between two candidates with the same dimension (knowing what is the dimension of the wave front set: see section III.4).

Let N be the set of all nilpotent elements of g. The following result gives us candidates for the wave front set.

Proposition III.1.4 (2.4 in [How81])

If G is a semisimple group and π an irreducible representation of G, then

It is known ([CM93] for example) that there are only finitely many conjugacy classes of nilpotent elements in g. So the same is true for the wave front sets.

One step is missing: how to compute the dimension of the wave front set ?

The dimension of the wave front set of a representation is twice its Gelfand-Kirillov dimension.

We will compute the Gelfand-Kirillov dimension using the following theorem.

Theorem III.1.2 (Theorem 1.2 [Vog78])

Suppose π ∈ Ĝ. Let X be the space of K-finite vectors in π, and let d be the Gelfand-Kirillov dimension of X. Let Ω be the Casimir operator of K. We write N X (t) for the sum of the dimensions of the eigenspaces of Ω with eigenvalue less than or equal to t 2 . Then there is a constant A > 0 depending on X, such that for t > 1,

III.2 Nilpotent elements of g

The nilpotent set N of g is the set of all nilpotent elements. Here g is of real rank one. So

We know where we can find the generators of the nilpotent orbits. But how many orbit are there and what are their dimension?

The following lemma gives us the different nilpotent orbits in the nilradical of g.

Lemma III.5.2

There are three nilpotent orbits in sp(n, 1):

+

The first one corresponds to the zero orbit. The second is the only one which is nilpotent of degree 2. It corresponds to the orbit generated by any element of g α/2 . The last one is the only one nilpotent of degree 1. So it is generated by any element of g α . The dimensions are given by Corollary 6.1.4 in [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF]. Also here we have to recall that the partition of the complex nilpotent orbit coming from a real nilpotent orbit is given by the boxes of the corresponding Young diagram.

The combination of the two lemmas is sufficient to conclude the results about wave front set of E k in Theorem 2 for Sp(n, 1). Proof. As in the complex case, one can compute the eigenvalue on each K-type τ p,q in H λ k α for the Casimir operator Ω k of k (using for example [Hal15, Proposition 10.6]). Recall from II.2.6 that τ p,q has highest weight µ p,q = p 2 1 + q 2 ( 2 + 3 + 4 ). Then one finds:

The dimension of the space of τ p,q follows from the Weyl dimension formula [Kna02, Theorem 5.84]:

where δ(p, q) := ( p+q

( p-q 2 )-1)! and C(q) := (q+1)(q+3)(q+5)(2q+8)(2q+6)(2q+4) 2 12 3 4 5 2 7

. For a fixed q, the sum of d p,q between 2 values p min and p max of p is telescopic:

The inequality given by the eigenvalue of the Casimir operator (III.21) bounds a region of K-types by an ellipsis. As in the previous case, we need to know the behaviour in t of the sum limited by this ellipsis when the ellipsis goes to infinity. The method to compute the N E k (t) of the [Vog78, Theorem 1.2] is exactly the same as the SU(n, 1) case and is left to the reader.

The following lemma gives us the different nilpotent orbits in the nilradical of g.

Lemma III.6.2

There are three nilpotent orbits in g:

1 the trivial orbit, 2 the orbit generated by g α/2 , of dimension 30, 3 the orbit generated by g α , of dimension 22.

Proof. The table on page 151 in [START_REF] Collingwood | Nilpotent orbits in semisimple Lie algebras[END_REF] ensures us that there are three possible Dynkin diagrams corresponding to the nilpotent orbits. Relating these diagrams to the table on page 128 allows us to find the dimensions associated to these (real) nilpotent orbit. Now we want to relate the orbits generated respectively by g α and g α/2 and the two nilpotent orbits of g. First choose an element X α/2 in g α/2 such that B X α/2 , θ(X α/2 ) = 0. Lemma 3.2 in [Hel01, Chapter IX] proves that for all X α ∈ g α ,

This proves that

The Jacobi identity allows us to conclude that Z(g α/2 ) ⊂ Z(g α ). This proves that the (nilpotent) orbit generated by g α has lower dimension than that of g α/2 .

The combination of the two lemmas above proves the results for F 4 about wave front set of E k in Theorem 2.

Remark

The above formula agrees with the Plancherel density for the real hyperbolic space for τ = triv K (and the, σ = triv M ). See e.g. [START_REF] Hilgert | Resonances and residue operators for symmetric spaces of rank one[END_REF]. It also gives the different irreducible continuous constituents of the Plancherel density for the p-forms on the real hyperbolic space, as computed in [START_REF] Pedon | Analyse harmonique des formes différentielles sur l'espace hyperbolique réel[END_REF].

A.2 Case of SU(n, 1)

Here G/K is the complex hyperbolic space. Recall that K is S(U(n) × U(1)) and

The elliptic Cartan subalgebra h C of g C consists of the diagonal matrices in sl(n+1, C).

Let { j } j=1,...,n be the elementary weights defined by j (H) = h j where H = diag(h 1 , h 2 , . . . , h n+1 ) described the elements of h C . We denote S, S + and S 0 respec- tively the set of roots, positive roots and simple roots of (g C , h C ). We index the sets of roots by the Lie algebra, to which they correspond.

As in [BS79, Lemma 4.2], the fixed (τ, V τ ) ∈ K has highest weight of the form

where a 1 ≥ . . . ≥ a n-1 ≥ a n and a i ∈ Z for all i = 1, . . . , n + 1.

Let σ ∈ M (τ ). By [BS79, Theorem 4.4]. The highest weight µ σ of σ ∈ M (τ ) has the form

where for all j, we have

The fact that we are working with zero trace matrices implies that n+1 j=1 j = 0. Thus we can relate to Miatello's form for the weights. We get the same formula as in [Mia79, p. 258]:

Miatello gives this following Plancherel density:

We recall that λ α is the complex number associated to λ ∈ a * C by (I.4).

A.3 Case of Sp(n, 1)

Here G/K is the quaternionic hyperbolic space. In this case, K is Sp(n) × Sp(1) and

Let us recall some Lie algebraic structure. The Lie algebra of G is g = sp(n, 1). Its

Let t denote the set of diagonal matrices in g. So t C is the elliptic Cartan subalgebra of g C . Let h -be a maximal abelian algebra of m. Then h := h -+ a is a Cartan subalgebra of g, and h C is a Cartan subalgebra of g C .

Let { j } j=1,...,n be the elementary weights defined by j (H) = h j , where H = diag(h 1 , h 2 , . . . , h n+1 ) ∈ t C . We denote S, S + and S 0 respectively the set of roots, positive roots and simple roots of (g C , h C ). We index the sets of roots by the Lie algebra, to which they correspond.

The fixed (τ, V τ ) ∈ K has highest weight of the form:

where a 1 ≥ . . . ≥ a n-1 ≥ a n ≥ 0, a n+1 ≥ 0 and a i ∈ Z for all i = 1, . . . , n + 1.

See [BS79, Lemma 5.2]. Theorem 5.5 in [START_REF] Baldoni | Branching theorems for semisimple lie groups of real rank one[END_REF] gives us the form of the highest weight µ σ of σ ∈ M (τ ) as follows:

where we have:

1 a j ≥ b j+1 , for all j = 1, . . . , n -1 A.4. Case of F 4 2 b j ≥ a j+1 , for all j = 2, . . . , n -1

, for some j = 0, . . . , min(a n+1 , b 1 ) and b 1 satisfies b 1 ∈ Z + and n j=1 (a j -b j ) ∈ 2Z. We want to use the Plancherel formula given by Miatello [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF]. But the Cartan subalgebra h C used in this thesis is not the same as the Cartan algebra t C used by Baldoni-Silva for the branching rules. So we have to use a Cayley transform to find the highest weight of σ relative to h C . This computation was already done in [BSK80].

Namely (see [START_REF] Helgason | Differential geometry, Lie groups, and symmetric spaces[END_REF]) one knows that for each β ∈ S we can select a root vector X β such that B(X a , X -a ) = 2 β,β and θX β = -X β . Choose β = 1 -n+1 . Then if we denote by e 1 , . . . , e n+1 the fundamental weights of the root system ∆(g C , h C ), we have :

So µ σ is written in the root system ∆(g C , h C ) as:

and [START_REF] Miatello | On the plancherel measure for linear lie groups of rank one[END_REF] gives the following Plancherel density:

We recall that λ α is the complex number associated to λ ∈ a * C by (I.4).

A.4 Case of F 4

Here, G = F 4 . We recall g = f -20

4

, K is Spin(9) and its Lie algebra is k = so(9). Let t ⊂ k be the compact Cartan subalgebra for both g and k. The branching rules for the fixed τ are determined in [BS79, Paragraph 6]. For a maximal abelian subspace a of p its centralizer in k is m = so(7). The problem is that this Lie algebra m is not contained in the standard way in k. So, we cannot use twice the branching rules for SO(n) directly. Let K 1 = Spin(8) a subgroup of K contained in the standard way. We denote by k 1 its Lie algebra.

Let { j } j=1,...,n be the elementary weights in ∆(g C , t C ). We denote S, S + and S 0 respectively the set of roots, positive roots and simple roots of (g C , h C ). We index the sets of roots by the Lie algebra, to which they correspond.

A.4. Case of F

Let H α 4 be the unique element of a such that α 4 = B(H α 4 , • ). Choose the root vectors X α 4 and X -α 4 such that [X α 4 , X -α 4 ] = H α 4 and X α 4 + X -α 4 ∈ p. Define a to be the one-dimensional space spanned by X α 4 + X -α 4 . Let h = h -⊕ a where h -is a Cartan subalgebra of m. Then the Cayley transform Ad exp π 4 (X α 4 -X -α 4 ) maps t C onto h C . The set of roots ∆(g C , h C ) is the following root system on m

( 1 -2 + 3 + 4 ) As said before, m is not contained in the standard way in k 1 . Let φ be the automorphism of k 1 which keeps the roots and such that φ(M ) is contained in the standard way in φ(K 1 ). This automorphism is given by

The fixed K-type τ has highest weight of the form:

where a 1 ≥ . . . ≥ a 4 ≥ 0, 2a i ∈ Z and a i -a j ∈ Z for all i, j = 1, . . . , 4.

Because of the branching rules of Spin(n), the representations of K 1 which are contained in the restriction of τ to K 1 have the following highest weights:

where a 1 ≥ c 1 . . . ≥ a 4 ≥ | c 4 | , 2c i ∈ Z and a i -c j ∈ Z for all i, j = 1, . . . , 4.

We have now to apply φ to this highest weight to use the branching rules of so(n). We get a highest weight of the form