J Muller

Cohomology of Deligne-Lusztig varieties associated to PEL Rapoport-Zink spaces with

Rapoport-Zink spaces are moduli spaces classifying the deformations of some pdivisible group equipped with additional structures. Its cohomology is expected to play a role in the local Langlands correspondences. In general, it is difficult to compute it especially outside of the supercuspidal part, which is described by the Kottwitz conjecture and which is known in a variety of cases. However, a certain small family of Rapoport-Zink spaces admit a Bruhat-Tits stratification on their special fiber, such that the strata are Deligne-Lusztig varieties of Coxeter type. It is the case in particular of the unitary PEL Rapoport-Zink spaces of signature p1, n ´1q with p inert or ramified. The closure of a Bruhat-Tits stratum is a generalized Deligne-Lusztig variety associated to a finite unitary or symplectic group. In the inert case, we compute the cohomology of an individual stratum entirely, and in the ramified case we describe a substantial part of it. Hyperspecial level in the inert case guarantees the triviality of the nearby cycles, allowing us to carry our computations to the analytical tubes of the closed Bruhat-Tits strata. These tubes form an open cover of the generic fiber of the Rapoport-Zink space, inducing a Čech spectral sequence which computes its cohomology. Exploiting this sequence, we prove that the cohomology of this Rapoport-Zink space in the inert case fails to be admissible in general. Eventually, via p-adic uniformization the cohomology of the Rapoport-Zink space is related to the cohomology of the supersingular locus of the associated PEL Shimura variety at hyperspecial level. For low values of n, we compute the cohomology of the supersingular locus through this sequence.

1 Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q I am first and foremost grateful to both my PhD co-supervisors, Pascal Boyer and Naoki Imai, whose guidance throughout the years has been a constant encouragement to keep going forward. I also wish to adress special thanks to Olivier Dudas who brought me very precious help on multiple occasions to understand the field of Deligne-Lusztig theory. I am thankful to Jean-Loup Waldspurger as well, who helped me understand the properties of compactly induced representations of locally profinite groups. Many thanks also to Torsten Wedhorn and Olivier Dudas for devoting so much time to be rapporteurs of my PhD thesis. I'd like to extend my thanks to Jean-François Dat, Sophie Morel and Jacques Tilouine as well for doing me an honor as members of the jury.

To Yuma and Léo

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q The Langlands program is a significant collection of conjectures, expected to unveil deep and unifying connections between different fields of mathematics. In particular, it predicts the existence of a correspondence between Galois representations and automorphic or smooth representations of reductive groups over a global or a local field. The first formulations of this research project date back to Langlands in 1967, giving a far-reaching generalization of Harish-Chandra and Gelfand's philosophy of cusp forms. It offered exciting new perspectives to the field of number theory, and ever since several dozen researchers contributed to expanding, deepening and maturing Langlands' intuition.

One approach to tackle Langlands conjectures is based on geometry. It is commonly expected that the correspondence one seeks to establish should be encrypted in the properties of geometric objects, known as Shimura varieties (global case) and Rapoport-Zink spaces (local case).

In the PEL case, these are moduli spaces respectively for abelian varieties or for p-divisible groups with extra structures. These symmetric spaces are equipped with actions of reductive groups over a global or a local field, so that their cohomology is expected to give a geometric incarnation of the conjectural Langlands correspondences. Different techniques have been used by various researchers to access the cohomology in some specific cases. In particular for the local case, the Kottwitz conjecture describes the supercuspidal part of the cohomology of Rapoport-Zink spaces. It has been proved in the Lubin-Tate case by Boyer [START_REF] Boyer | Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale[END_REF] and by Harris and Taylor [START_REF] Harris | The Geometry and Cohomology of Some Simple Shimura Varieties[END_REF]. It was established for all unramified Rapoport-Zink spaces of EL type by Fargues [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] and Shin [START_REF] Shin | On the cohomology of Rapoport-Zink spaces of EL-type[END_REF]. Eventually, it was proved for the unramified unitary Rapoport-Zink spaces of PEL type in an odd number of variables by Nguyen and Bertoloni-Meli [START_REF] Nguyen | Un cas PEL de la conjecture de Kottwitz[END_REF] and [START_REF] Bertoloni Meli | The Kottwitz conjecture for unitary PELtype Rapoport-Zink spaces[END_REF]. The non-supercuspidal part is more difficult to grasp, and there is no conjecture to describe it. So far, it has only been computed in the Lubin-Tate case by Boyer [START_REF] Boyer | Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples[END_REF], and the case of the Drinfeld space then followed by duality (see for instance [START_REF] Fargues | L'isomorphisme Entre Les Tours de Lubin-Tate et de Drinfeld[END_REF]). The specific geometry of the Lubin-Tate case allowed for explicit computations, however the same approach does not apply to more general cases, so that the non-supercuspidal part seems to be currently out of reach.

There exists however a certain small family of Rapoport-Zink spaces whose special fiber exhibits some very nice geometric properties. Such spaces are said to be "fully Hodge-Newton decomposable" and they have been fully classified by Görtz, He and Nie in [START_REF] Görtz | Fully Hodge Newton Decomposable Shimura Varieties[END_REF] using a group theoretic approach. The special fiber of a fully Hodge-Newton decomposable Rapoport-Zink space admits a stratification by Deligne-Lusztig varieties, and the incidence relations of the stratification is closely related to the combinatorics of the Bruhat-Tits building of an underlying p-adic group. Consequently, this stratification is known as the Bruhat-Tits stratification. The Rapoport-Zink space is said to be "of Coxeter type" if it is fully Hodge-Newton decomposable, and if the Deligne-Lusztig varieties occuring in the Bruhat-Tits stratification are of Coxeter

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

type. This subfamily of Rapoport-Zink space has also been entirely classified by Görtz, He and Nie in their subsequent work [START_REF] Görtz | Basic loci of Coxeter type with arbitrary parahoric level[END_REF].

To our knowledge, the first time that Deligne-Lusztig varieties were explicitely mentioned in the context of the Langlands program was in [START_REF] Yoshida | On Non-Abelian Lubin-Tate Theory via Vanishing Cycles[END_REF], dealing with the Lubin-Tate tower. However, it is the pioneering work of Vollaard and Vollaard-Wedhorn in [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] and [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] which coined the notion of Bruhat-Tits stratification. The authors used an approach based on Dieudonné theory and the combinatorics of vertex lattices in a hermitian space. The corresponding space was the GUp1, n ´1q PEL Rapoport-Zink space at inert p and hyperspecial level. Mimicking their approach, Rapoport, Terstiege and Wilson dealt with the case of GUp1, n ´1q at a ramified p and level given by a selfdual lattice in [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF]. This paved the way to the study of the geometry of the special fiber on a case-by-case basis by several authors, using either a similar Dieudonné theoretic approach or a group theoretic approach:

-the case of GUp2, 2q at inert p and hyperspecial level by Howard and Pappas in [START_REF] Howard | On the Supersingular Locus of the GU(2,2) Shimura Variety[END_REF], and by Wang in [START_REF] Wang | On the Bruhat-Tits Stratification for GU(2,2) Type Rapoport-Zink Space: Unramified Case[END_REF] using another method which also covers the case of split p, -the case of GUp1, n ´1q at ramified p and parahoric level of exotic good reduction by Wu in [START_REF] Wu | The Supersingular Locus of Unitary Shimura Varieties with Exotic Good Reduction[END_REF], -the case of spinor groups GSpinpn, 2q at hyperspecial level by Howard and Pappas in [START_REF] Howard | Rapoport-Zink Spaces for Spinor Groups[END_REF], -the case of GpUp1, n ´1q ˆUp1, n ´1qq at unramified p and hyperspecial level by Helm, Tian and Xiao in [START_REF] Helm | Tate Cycles on Some Unitary Shimura Varieties Mod p[END_REF], -the case of GUp1, n ´1q at inert p and arbitrary maximal parahoric level by Cho in [START_REF] Cho | The Basic Locus of the Unitary Shimura Variety with Parahoric Level Structure, and Special Cycles[END_REF], -the case of spinor groups GSpinpn, 2q at certain non-hyperspecial level by Oki in [START_REF] Oki | Rapoport-Zink Spaces for Spinor Groups with Special Maximal Parahoric Level Structure[END_REF], -the case of a quaternionic unitary space at parahoric level by Wang in [START_REF] Wang | On the Bruhat-Tits Stratification of a Quaternionic Unitary Rapoport-Zink Space[END_REF] and [START_REF] Wang | On Quaternionic Unitary Rapoport-Zink Spaces with Parahoric Level Structures[END_REF], and independently at maximal special parahoric level by Oki in [START_REF] Oki | On Supersingular Loci of Shimura Varieties for Quaternionic Unitary Groups of Degree 2[END_REF], -the case of GUp2, 2q at ramified p and at special maximal parahoric level by Oki in [START_REF] Oki | On the Supersingular Locus of the Shimura Variety for GU(2,2) over a Ramified Prime[END_REF], -the case of GUp2, n ´2q at inert p and hyperspecial level by Fox and Imai in [START_REF] Fox | The Supersingular Locus of the Shimura Variety of GU(2,n-2)[END_REF], -the case of GLp4q as well as GUp2, 2q at split p and hyperspecial level by Fox in [START_REF] Fox | The GL4 Rapoport-Zink Space[END_REF].

Aside from the cases studied in [START_REF] Cho | The Basic Locus of the Unitary Shimura Variety with Parahoric Level Structure, and Special Cycles[END_REF] and in [START_REF] Fox | The Supersingular Locus of the Shimura Variety of GU(2,n-2)[END_REF], all the Rapoport-Zink spaces cited above are of Coxeter type. The spaces of [START_REF] Cho | The Basic Locus of the Unitary Shimura Variety with Parahoric Level Structure, and Special Cycles[END_REF] are fully Hodge-Newton decomposable but not of Coxeter type when the parahoric level is not special, and the space of [START_REF] Fox | The Supersingular Locus of the Shimura Variety of GU(2,n-2)[END_REF] is not fully Hodge-Newton decomposable when n ě 5. In particular, the strata which are built in loc. cit. are not necessarily Deligne-Lusztig varieties.

Deligne-Lusztig varieties naturally arise in Deligne-Lusztig theory, a field of mathematics whose aim is the classification of all irreducible complex representations of finite groups of Lie type, ie. reductive groups over finite fields. Let G be a connected reductive group over an algebraic closure F p of F p . Let q be a power of p and assume that G has an F q -structure, induced by a Frobenius morphism F : G Ñ G. Let G :" GpF q q » G F be the associated finite group of Lie type. A Levi complement L Ă G is the group of F q -points of some rational Levi complement L of G. Such a Levi complement L is said to be split if L is the Levi complement of a rational parabolic subgroup P of G. One way of building irreducible representations of G is Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q to decompose representations parabolically induced from proper split Levi complements L of G. However, this process fails to recover the cuspidal representations. To remedy this issue, Deligne and Lusztig defined in their innovative work [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF] new induction functors from any (not necessarily split) Levi L of G, generalizing the usual parabolic induction. They did so by associating a certain variety Y LĂP to any parabolic subgroup P of G with rational Levi complement L, which is naturally equipped with commuting actions of G and of L " L F . The alternate sum of the cohomology of Y LĂP provides a virtual G´bimodule´L, which is used to define the Deligne-Lusztig induction functor R G L between the categories of representations of L and of G. Reducing to the case where L " T is a maximal torus in G and computing explicitely the decompositions of the induced representations R G T θ for all characters θ of T , Lusztig managed in [START_REF] Lusztig | Characters of Reductive Groups over a Finite Field[END_REF] to give a complete classification of all irreducible representations of all simple finite groups of Lie type.

To sum up, the geometry of certain Rapoport-Zink spaces can be described in terms of Deligne-Lusztig varieties, and cohomology plays a crucial role in both the Langlands program and Deligne-Lusztig theory. This observation is the starting point of this PhD thesis, whose aim is to derive the consequences of the geometric connections established by the authors cited above for the cohomology. We brought our attention to the two cases that have been chronologically first considered, that is the unitary PEL Rapoport-Zink space of signature p1, n ´1q over a prime p which is inert, see [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] and [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF], or ramified, see [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF].

In the following, we first detail the general approach before stating the results reached in the inert case. Eventually, we explain how we plan to adapt the method to the ramified case.

1.2 The case of inert or ramified PEL unitary Rapoport-Zink space of signature p1, n ´1q

If E is a p-adic field where p ą 2, let O E denote its ring of integers,let π denote a uniformizer and let κpEq be the residue field. Let Nilp E be the category of O E -schemes where π is locally nilpotent. Assume now that E{Q p is quadratic and denote by ¨the non-trivial element of GalpE{Q p q. If E{Q p is ramified, we may chose π so that π " ´π. If E{Q p is unramified, then E » Q p 2 :" W pF p 2 q Q , and O E » Z p 2 :" W pF p 2 q, where W p¨q denotes the ring of Witt vectors.

Let E 1 {E be an unramified extension. For S P Nilp E 1 , a unitary p-divisible group of signature p1, n ´1q over S is a triple pX, ι X , λ X q such that -X is a p-divisible group over S, ι X : O E Ñ EndpXq is a O E -action on X such that the induced action on its Lie algebra satisfies the Kottwitz signature p1, n ´1q condition: @a P O E , charpιpaq | LiepXqq " pT ´aq 1 pT ´aq n´1 , -if E{Q p is ramified, then the induced action of ι X on LiepXq also satisfies the Pappas condition: λ X : X " Ý Ñ t X is a principal polarization, where t X denotes the Serre dual of X. We assume that the associated Rosati involution induces ¨on O E .

Note that charpιpaq | LiepXqq is a polynomial with coefficients in O S . The Kottwitz condition compares it with a polynomial with coefficients in O E Ă O E 1 via the structure morphism S Ñ O E 1 . Let us fix such a unitary p-divisible group pX, ι X , λ X q of signature p1, n´1q over κpE 1 q, such that X is superspecial. We call the triple pX, ι X , λ X q the framing object of the Rapoport-Zink space. If E{Q p is unramified or if E » Q p r ? ´ps then one may take E 1 " E. However if E » Q p r ? ǫps where ǫ P Z p is such that ´ǫ is not a square in Z p , then one must take E 1 " E b Qp W pF p 2 q Q in order to define the framing object. The Rapoport-Zink space is the moduli space M classifying the deformations of the framing object by quasi-isogenies. More precisely, for S P Nilp E 1 , MpSq is the set of isomorphism classes of tuples pX, ι X , λ X , ρ X q where pX, ι X , λ X q is a unitary p-divisible group of signature p1, n ´1q over S, and where ρ X : X ˆS S Ñ X ˆκpE 1 q S is an O E -linear quasi-isogeny such that t ρ X ˝λX ˝ρX " cλ X for some c P Q p . Here S is the special fiber of S and t ρ X is the dual quasi-isogeny. By the work of Rapoport and Zink in [START_REF] Rapoport | Period Spaces for "p"-divisible Groups (AM-141)[END_REF] and of Pappas in the ramified case, the functor M is a formal scheme over SpfpO E 1 q which is formally of finite type, formally smooth in the inert case, and flat in the ramified case.

Remark. In the inert case and in the ramified case with n odd, any choice of the framing object X gives the same Rapoport-Zink space. In the ramified case with n even however, there are essentially two choices of framing objects, giving rise to two different spaces. These two cases are refered to as the split and non-split cases, see [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF] Remark 4.2.

Let M red denote the special fiber of M. The Bruhat-Tits stratification, which is built in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] for the inert case and in [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF] for the ramified case, can be written as

M red " ğ ΛPL M Λ,
where each stratum M Λ is a locally closed subvariety which is defined over κpEq, and Λ runs over the set L of so-called vertex lattices. More precisely, the indices Λ are almost self-dual O E -lattices in a certain E{Q p -hermitian space of dimension n (denoted by V in the inert case, and by C in the ramified case). Let J denote the group of unitary similitudes of this hermitian space, so that J acts on the set of vertex lattices L. The group J can also be identified with the group AutpX, ι X , λ X q of automorphisms of the framing object. In particular, J also acts on M by g ¨pX, ι X , λ X , ρ X q :" pX, ι X , λ X , g ˝ρX q for all g P J.

For all Λ P L let M Λ denote the closure of the stratum M Λ. The J-action on the special fiber M red is compatible with the Bruhat-Tits stratification, in the sense that any g P J induces an isomorphism g :

M Λ " Ý Ñ M gpΛq ,
and thus an isomorphism between the closed strata M Λ " Ý Ñ M gpΛq as well. Let J Λ :" Fix J pΛq be the fixator in J of Λ P L. Then J Λ is a maximal compact subgroup of J, and it admits a finite quotient which is isomorphic to a finite group of unitary similitudes GU tpΛq pF p q in the inert case, and to a finite group of symplectic similitudes GSp tpλq pF p q is the ramified case. Here Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q 0 ď tpΛq ď n is a certain integer called the orbit type of Λ P L, and which is odd in the inert case and even in the ramified case. It turns out that the induced action of J Λ on M Λ factors through an action of this finite quotient. The Bruhat-Tits stratification is very well behaved for the two following reasons.

(1) The set L of vertex lattices can be given the structure of a polysimplicial complex, whose combinatorics describes the incidence relations between the closed Bruhat-Tits strata.

(2) Each closed Bruhat-Tits stratum, equipped with its action of the finite quotient of the maximal compact subgroup J Λ , is naturally isomorphic to a generalized Deligne-Lusztig variety for GU tpΛq pF p q in the inert case, and for GSp tpΛq pF p q in the ramified case.

The polysimplicial complex L of (1) is closely related to the Bruhat-Tits building BT of J over Q p . In fact, both polysimplicial complexes are equal except in the case of split ramified p with n even. There, L is a slight modification of BT, see [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF] Proposition 3.4. The isomorphism of (2) also induces an isomorphism between M Λ and the Coxeter variety for GU tpΛq pF p q or for GSp tpΛq pF p q. This is in accordance with the fact that the Rapoport-Zink space M is of Coxeter type in both the inert and the ramified cases, by [START_REF] Görtz | Basic loci of Coxeter type with arbitrary parahoric level[END_REF].

Let M an denote the generic fiber of the formal scheme M in the sense of Berkovich. Thus, M an is a smooth analytical space of dimension n´1 over E 1 . Let red : M an Ñ M red denote the reduction map. It is anticontinuous, ie. the preimage of a closed (resp. open) subset is open (resp. closed). In order to derive the consequences of such a stratification for the cohomology, we establish the following strategy.

(A) Understand the cohomology H ' c pM Λ b F p , Q ℓ q of an individual closed Bruhat-Tits stratum by using Deligne-Lusztig theory via (2).

(B) Introduce the analytical tubes U Λ :" red ´1pM Λ q, and study the cohomology of the Rapoport-Zink space M an via the Čech spectral sequence associated to the open cover tU Λ u Λ , whose combinatorics is described by (1).

Remark. By general theory, there is a connected reductive group G over Q p , a parahoric subgroup K 0 Ă GpQ p q, and a finite étale cover M K Ñ M an for every open compact subgroup K Ă K 0 . Here, G is the group of unitary similitudes of some n-dimensional E{Q p -hermitian space, and J is an inner form of G. Moreover the group K 0 is maximal special, and in the inert case it is hyperspecial. For K Ă K 1 , there are transition maps Π K,K 1 : M K 1 Ñ M K so that the spaces M K fit together in a projective system M 8 :" pM K q K , called the Rapoport-Zink tower. The action of J on M can be extended to an action on each M K which is compatible with the transition maps. Therefore M 8 is equipped with an action of GpQ p qˆJ, where GpQ p q acts on the structure level by Hecke correspondences. One may define the cohomology of M 8 via the formula H ' c pM 8 , Q ℓ q :" lim Ý Ñ

K lim Ý Ñ U K lim Ð Ý k H ' c pU K p b C p , Z{ℓ k Zq b Q ℓ ,
where U K runs over all the relatively compact open subsets of M K . These cohomology groups are representations of GpQ p q ˆJ ˆW , where W " W E denotes the absolute Weil group of E. We note that the action of W E is induced by Rapoport and Zink's (non effective) Weil descent datum on M, as defined in [START_REF] Rapoport | Period Spaces for "p"-divisible Groups (AM-141)[END_REF] 3.48. These cohomology groups are the main object of interest in the context of the Langlands program. In this thesis, our results only deal with the cohomology of the generic fiber M an " M K 0 . Thus, we focus on the cohomology groups H ' c pM an , Q ℓ q " H ' c pM 8 , Q ℓ q K 0 , as representations of J ˆW . The reason is that our approach is fruitful only in situations where one may relate the cohomology of a closed Bruhat-Tits stratum M Λ in the special fiber, to the cohomology of its analytical tube U Λ,K :" Π ´1 K,K 0 red ´1pM Λ q in M K . In the inert case, this can be achieved trivially when K " K 0 since the formal smoothness of M insures the triviality of the nearby cycles. In future works, we hope to generalize our approach to more general parahoric level structures, in which case the semi-stable reduction should allow explicit computations of the nearby cycles. Other cases of bad reduction, such as the ramified case, may also be manageable as we discuss in the end of the introduction.

Step (A): the cohomology of an individual closed Bruhat-Tits stratum

As mentioned in (2), each closed Bruhat-Tits stratum is isomorphic to a generalized Deligne-Lusztig variety. Let us explain what we mean by this. In general, let G be a connected reductive group over F p equipped with a Frobenius morphism F : G Ñ G inducing an F q -structure. Let G :" G F be the associated finite group of Lie type. Let P be a parabolic subgroup of G. The associated generalized Deligne-Lusztig variety is X P :" tgP P G{P | g ´1F pgq P PF pPqu.

It is defined over F q δ where δ ě 1 is the smallest integer such that F δ pPq " P, and it is equipped with an action G ñ X P by left translations. We say that a generalized Deligne-Lusztig variety X P is classical if in addition, there exists a rational Levi complement L Ă P. When this condition is satisfied, the Deligne-Lusztig variety inherits an action X P ð L :" L F by right translations, which commutes with the action of G. In this case, the cohomology of X P is a G-bimodule-L, and can be used to defined the Deligne-Lusztig induction functor between the categories of representations of L and of G. We note that the varieties denoted above by Y LĂP are in fact some L-torsor of X P . Thus, in the context of Deligne-Lusztig theory which focuses on the study of the induction functors afforded by the varieties Y LĂP , one is only interested in classical Deligne-Lusztig varieties.

For this reason, to our knowledge their generalized versions have not been systematically studied in the literature, except in [START_REF] Bonnafé | On the irreducibility of Deligne-Lusztig varieties[END_REF] where a criterion for the irreducibility of X P is proved. It turns out that the Deligne-Lusztig variety to which a closed Bruhat-Tits stratum M Λ is isomorphic, is not classical. Therefore, no result regarding its cohomology can be directly read from the literature.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

However, the works of [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] in the inert case, and of [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF] in the ramified case, give us enough geometric understanding of M Λ in order to access its cohomology. Let us write tpΛq " 2θ `1 in the inert case, and tpΛq " 2θ in the ramified case. There exists a stratification (called the Ekedahl-Oort stratification in the inert case)

M Λ " ğ 0ďθ 1 ďθ M Λ pθ 1 q,
where each M Λ pθ 1 q is a locally closed subvariety, and the closure of the stratum associated to θ 1 is the union of the all the strata associated to t ď θ 1 . The isomorphism of (2) between M Λ and a generalized Deligne-Lusztig variety, naturally induces an isomorphism between M Λ pθ 1 q and a classical Deligne-Lusztig variety which is, in some sense, parabolically induced from the Coxeter variety for the smaller group of unitary similitudes GU 2θ 1 `1pF p q in the inert case, and for the smaller group of symplectic similitudes GSp 2θ 1 pF p q in the ramified case.

In [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF], Lusztig has computed the cohomology of the Coxeter varieties for all finite classical groups in terms of unipotent representations. The unipotent representations of GU 2θ`1 pF p q are classified by the integer partitions λ of 2θ `1 and we denote them ρ Λ , see 2.2.1. The unipotent representations of GSp 2θ pF p q are classified by Lusztig's notion of symbols S of rank θ and of odd defect, and we denote them ρ S . In 2.4.3 and 4.3.2, we translate Lusztig's results of [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] in terms of the classification by integer partitions or by symbols respectively. From [START_REF] Geck | The Character Theory of Finite Groups of Lie Type: A Guided Tour[END_REF] and [START_REF] Geck | Characters of Finite Coxeter Groups and Iwahori-Hecke Algebras[END_REF], we derive the combinatorical rules to compute parabolic induction of unipotent representations. It allows us to entirely determine the cohomology of a stratum M Λ pθ 1 q. Then, we study the spectral sequence associated to the stratification

E a,b 1 " H a`b c pM Λ paq b F p , Q ℓ q ùñ H a`b c pM Λ b F p , Q ℓ q,
which degenerates on the second page thanks to the repartition of the Frobenius eigenvalues throughout the sequence. The variety M Λ is projective of dimension θ, and it is smooth only in the inert case, and in the ramified case when θ ď 1. Let τ P GalpF p {κpEqq be the geometric Frobenius relative to κpEq.

In the inert case, the purity of the Frobenius on the cohomology of M Λ allows us to compute all the E 2 terms, and it leads to the following statement.

Theorem (2.5.1). In the inert case, let Λ P L and write tpΛq " 2θ `1.

(1) The cohomology group

H j c pM Λ b F p , Q ℓ q is zero unless 0 ď j ď 2θ.
(2) The Frobenius τ acts like multiplication by p´pq j on H j c pM Λ b F p , Q ℓ q.

(3) For 0 ď j ď θ we have

H 2j c pM Λ b F p , Q ℓ q " minpj,θ´jq à s"0 ρ p2θ`1´2s,2sq .
For 0 ď j ď θ ´1 we have

H 2j`1 c pM Λ b F p , Q ℓ q " minpj,θ´1´jq à s"0 ρ p2θ´2s,2s`1q .
In particular, all irreducible representations in the cohomology groups of even index belong to the unipotent principal series, whereas all the ones in the groups of odd index have cuspidal support determined by the unique cuspidal unipotent representation of GU 3 pF p q, which is denoted ρ ∆ 2 with ∆ 2 equal to the partition p2, 1q of 3. The cohomology group H j c pM Λ b F, Q ℓ q contains no cuspidal representation of GU 2θ`1 pF p q unless θ " j " 0 or θ " j " 1. If θ " 0 then H 0 c is the trivial representation of GU 1 pF p q " F p2 , and if θ " 1 then H 1 c is the representation ρ ∆ 2 of GU 3 pF p q.

In the ramified case, unless when θ " 0 or 1, in which case M Λ is respectively isomorphic to a point or to P 1 , we do not have a full understanding of the cohomology of M Λ , but we can still get substantial information from the spectral sequence. For 0 ď k ď 2θ the weights of the Frobenius τ on the cohomology group H k c pM Λ b F p , Q ℓ q form a subset of tp i , ´pj`1 u for k ´minpk, θq ď i ď k ´rk{2s and for k ´minpk, θq ď j ď k ´rk{2s ´1. Among other things, if i, j ą k ´minpk, θq then we determine the eigenspaces of the Frobenius H

k c pM Λ b F p , Q ℓ q p i and H k c pM Λ b F p , Q ℓ q ´pj`1
explicitely up to at most four irreducible representations of GSp 2θ pF p q. We refer to 4.4.2 and 4.4.3 for the detailed results, as it would be too long to fit this introduction. In particular, we note that the action of the Frobenius on the cohomology is not pure when θ ě 3 (for θ " 2 the non-purity is undetermined). This is in accordance with M Λ not being smooth for θ ě 2. All irreducible representations of GSpp2θ, F p q occuring in an eigenspace of τ for an eigenvalue of the form p i belong to the unipotent principal series, whereas those corresponding to an eigenvalue of the form ´pj`1 belong to the cuspidal series determined by the unique cuspidal unipotent representation of GSpp4, F p q which is denoted by ρ S 2 , where S 2 is the symbol defined in 4.2.5. We note in particular that H k c pM Λ b F p , Q ℓ q contains no cuspidal representation of GSp 2θ pF p q, unless θ " k " 0 or θ " k " 2. When θ " 0 then H 0 c is the trivial representation of GSp 0 pF p q » t1u, and when θ " 2 then the eigenspace of τ in H 2 c for the eigenvalue ´p is ρ S 2 .

Remark. We observe that in the inert case, the non-principal cuspidal series determined above contributes to the cohomology of M Λ for θ ě 1, but in the ramified case it only contributes for θ ě 2. Moreover, in the inert case the two cuspidal series contribute separately to groups of even or odd degrees, but in the ramified case both series contribute to cohomology groups of degrees of any parity.

Step (B): on the cohomology of the inert Rapoport-Zink space at hyperspecial level

From now on, we consider only the inert case. Let L max denote the subset of all vertex lattices Λ P L having maximal orbit type tpΛq " t max . We have

t max " $ & % n if n is odd, n ´1 if n is even.
Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

Let us write t max " 2θ max `1. Then tU Λ u ΛPL max forms an open cover of the generic fiber M an to which one can associate the following J ˆW -equivariant Čech spectral sequence, concentrated in degrees a ď 0 and 0 ď b ď 2pn ´1q,

E a,b 1 : à γPI ´a`1 H b c pU pγq, Q ℓ q ùñ H a`b c pM an , Q ℓ q.
See 3.4.1.4 for some details regarding the definition of the W -action. Here, for a ď 0 the set I ´a`1 is defined by

I ´a`1 :" # γ " pΛ 1 , . . . , Λ ´a`1 q ˇˇˇˇ@
1 ď j ď ´a `1, Λ j P L pmq and U pγq :" ´a`1 č j"1

U Λ j " H + .
By the properties of the Bruhat-Tits stratification, if γ P I ´a`1 then there exists a unique vertex lattice Λpγq P L such that U pγq " U Λpγq . Thus, we must first relate the cohomology of any U Λ with the cohomology of its special fiber M Λ that we have investigated in step (A). Each cohomology group H b c pU pγq, Q ℓ q is naturally a representation of pJ Λ ˆIqτ Z where I Ă W is the inertia subgroup, and τ :" pp ´1 ¨id, Frobq P J ˆW is called the rational Frobenius element. Here, Frob P W is a fixed lift of the geometric Frobenius, and p ´1 ¨id is seen as an element of the center ZpJq » Q p2 (recall that J is a group of unitary similitudes).

Proposition (3.4.1.5). Let Λ P L and let 0 ď b ď 2pn ´1q. Write tpΛq " 2θ `1. There is a natural pJ Λ ˆIqτ Z -equivariant isomorphism H b c pU Λ , Q ℓ q " Ý Ñ H b´2pn´1´θq c pM Λ b F p , Q ℓ qpn ´1 ´θq.
On the right-hand side the inertia I acts trivially, the rational Frobenius τ acts like the geometric Frobenius τ defined in step (A), and the J Λ -action factors through its finite unitary or symplectic similitudes quotient.

This proposition relies on the fact that we consider the Rapoport-Zink space M an " M K 0 at hyperspecial level, insuring the triviality of the nearby cycles between U Λ and M Λ . It follows that τ acts like multiplication by the scalar p´pq b on any term E a,b 1 . Thus, the spectral sequence degenerates on the second page and the filtration on the abutment splits, ie. the k-th cohomology group of M an is the direct sum of the E a,b 2 terms on the diagonal a `b " k, see 3.4.1.7. In order to study the J-action, we rewrite the terms E a,b 1 in terms of compact inductions. Let tΛ 0 , . . . , Λ θmax u be a maximal simplex in L such that for all θ, tpΛ θ q " 2θ `1. We write J θ :" J Λ θ . We also define K pθq ´a`1 :" tγ P I pθq ´a`1 | Λpγq " Λ θ u, which is a finite subset of I ´a`1 equipped with an action of J θ . Proposition (3.4.1.10). We have an equality

E a,b 1 " m à θ"0 c ´Ind J J θ ´Hb c pU Λ θ , Q ℓ q b Q ℓ rK pθq ´a`1 s
where

Q ℓ rK pθq ´a`1 s denotes the permutation representation associated to J θ ñ K pθq ´a`1 .
Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

By exploiting this spectral sequence, we are able to compute the cohomology groups of M an of highest degree 2pn ´1q. We denote by J ˝the subgroup of J generated by all the compact subgroups. It corresponds to all the unitary similitudes in J whose multipliers are a unit. We note that J ˝is normal in J with quotient J{J ˝» Z.

Proposition (3.4.1.12). There is an isomorphism

H 2pn´1q c pM an , Q ℓ q » c ´Ind J J ˝1,
and the rational Frobenius τ acts via multiplication by p 2pn´1q .

When θ max " 1 (ie. n " 3 or 4), the Bruhat-Tits building of J is essentially a tree. Exploiting its combinatorics and the spectral sequence, we are also able to compute the group of degree 2pn ´1q ´1. Recall the representation ρ ∆ 2 which we introduced in the previous section.

Theorem (3.4.3.4). Assume that θ max " 1. We have

H 2pn´1q´1 c pM an , Q ℓ q » c ´Ind J J 1 ρ ∆ 2 ,
with the rational Frobenius τ acting via multiplication by ´p2pn´1q´1 .

In general, the terms E a,b 2 in the second page may be difficult to compute. However, the terms corresponding to a " 0 and b P t2pn ´1 ´θmax q, 2pn ´1 ´θmax q `1u are not touched by any non-zero differential in the alternating version of the Čech spectral sequence, making their computations accessible. Proposition (3.4.1.11). We have an isomorphism of J-representations E 0,2pn´1´θmaxq 2 » c ´Ind J J θmax 1.

If n ě 3 then we also have an isomorphism

E 0,2pn´1´θmaxq`1 2 » c ´Ind J J θmax ρ p2θmax,1q .
Here 1 denotes the trivial representation, and ρ p2θmax,1q denotes (the inflation to J θmax of) the unipotent representation of GU 2θmax pF p q associated to the partition p2θ max , 1q.

The previous statement has important consequences for the cohomology of M an . To explain it, let us recall a certain property of compactly induced representations. Let χ be a continuous character of the center ZpJq » Q p2 and let V be a smooth representation of J. Let V χ be the maximal quotient of V on which ZpJq acts through χ. Let K be an open compact subgroup of J and let ρ be an irreducible smooth representation of K. Assume that χ agrees with the central character of ρ on ZpJq X K. Then

pc ´Ind J K ρq χ » c ´Ind J ZpJqK χ b ρ " V ρ,χ,0 ' V ρ,χ,8 ,
see 3.4.2.2 and 3.4.2.3. The decomposition on the right-hand side follows from a general theorem in [START_REF] Bushnell | Induced representations of locally profinite groups[END_REF]. The J-representation V ρ,χ,0 is the sum of all supercuspidal subrepresentations

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q of c ´Ind J ZpJqK χ b ρ. This is a finite sum. The space V ρ,χ,8 contains no non-zero admissible subrepresentation, in particular it contains no irreducible subrepresentation but it may admit many irreducible quotients and subquotients, none of which is supercuspidal. We note that V ρ,χ,0 or V ρ,χ,8 may be zero. Therefore, the behaviour of a compactly induced representation as above depends greatly on whether there exists some irreducible supercuspidal subquotient in c ´Ind J K ρ. The existence of such subquotients may be elucidated by type theory, especially in the case where ρ is inflated from a finite quotient of K. Combining with the two previous propositions, we deduce the following statements. Note that we consider unramified characters of ZpJq because any unipotent representation has trivial central character.

Proposition (3.4.2.12). Let χ be any unramified character of ZpJq » Q p2 .

-assume that n ě 3. The representation pE 0,2pn´1´θmaxq 2 q χ contains no non-zero admissible subrepresentation, and it is not J-semisimple. If n ě 5, then the same statement holds for pE 0,2pn´1´θmaxq`1 2 q χ . -for n " 1 (resp. n " 3, 4), let b " 0 (resp. b " 3, 5). Then pE 0,b 2 q χ is an irreducible supercuspidal representation of J. If n " 2, then pE 0,2 2 q χ is the sum of two non-isomorphic supercuspidal representations of J.

In particular, we obtain the following corollary.

Corollary. Let χ be any unramified character of ZpJq.

If n ě 3 then H 2pn´1´θmaxq c pM an , Q ℓ q χ is not J-admissible. If n ě 5 then H 2pn´1´θmaxq`1 c pM an , Q ℓ q χ is not J-admissible.
This non-admissibility result shows a different behaviour from the cases of the Lubin-Tate tower or of the Drinfeld space.

1.5 The cohomology of the supersingular locus of the associated Shimura variety at an inert prime for n " 3, 4

The Rapoport-Zink space M is related to the supersingular locus of a certain PEL Shimura variety via the p-adic uniformization theorem, and a certain spectral sequence relates the cohomology of both spaces. In particular, for small values of n, our results so far allow us to compute the cohomology of the supersingular locus both in the inert case. Let us give some more details.

Let E be an imaginary quadratic field, and let V be an n-dimensional non-degenerate E{Qhermitian space of signature p1, n ´1q at infinity, and such that V b Q p is isomorphic to the hermitian space defining the group of unitary similitudes G. In particular E p » Q p 2 , so that p is inert in E. Let G be the group of unitary similitudes of V, seen as a reductive group over Q. Then G Qp " G and G R " GUp1, n ´1q. Assume that there exists a self-dual O Elattice Γ in V, and let StabpΓq denote the compact subgroup of GpA f q of elements g such that

gpΓ b Z p Zq " Γ b Z p Z.
Here A f denotes the ring of finite adèles. For any open compact subgroup

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q K p Ă StabpΓq X GpA p f q which is small enough, there is an integral model S K p of the associated PEL Shimura variety which is defined over O E . Since we have hyperspecial level structure at p, the integral model S K p is smooth and quasi-projective. Let S K p denote the special fiber of S K p , and let S ss K p denote the supersingular locus. Let I be the inner form of G such that IpQ p q " J, I A p f " G A p f and I R " GUp0, nq. The p-adic uniformization theorem of [START_REF] Rapoport | Period Spaces for "p"-divisible Groups (AM-141)[END_REF] gives natural isomorphisms of analytic spaces over E 1

IpQqzpM an ˆGpA p f q{K p q " Ý Ñ p S ss,an K p b E E 1 ,
which are compatible as the level K p varies. Here p S ss,an

K p
denotes the analytical tube of the supersingular locus inside the analytification of the generic fiber of S K p . Associated to this geometric identity, Fargues has built in [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] a spectral sequence computing the cohomology of p S ss,an K p . Since S K p is smooth, it amounts to the cohomology of the supersingular locus S ss K p itself. The pGpA p f q ˆW q-equivariant spectral sequence takes the following shape

F a,b 2 " à ΠPA ξ pIq Ext a J `H2pn´1q´b c pM an , Q ℓ qp1 ´nq, Π p ˘b Π p ùñ H a`b c pS ss b F p , L ξ q,
where ξ is a finite dimensional irreducible algebraic Q ℓ -representation of G of weight wpξq P Z, L ξ is the associated local system on the Shimura variety S K p , A ξ pIq is the space of all automorphic representations of IpAq of type q ξ at infinity, and

H ' c pS ss b F p , L ξ q :" lim Ý ÑK p H ' c pS ss K p b F p , L ξ q. By [Far04] Lemme 4.4.12, we have F a,b
2 " 0 as soon as a is strictly bigger than the semisimple rank of J, which is equal to θ max . In particular, if θ max ď 1 then all the differentials are zero and the spectral sequence is already degenerated, allowing us to compute the abutment entirely. Since the case θ max " 0 is kind of trivial, we now assume θ max " 1 (ie. n " 3 or 4). In particular, the supersingular locus S ss K p has dimension θ max " 1. Let X un pJq denote the set of unramified characters of J. Let St J denote the Steinberg representation of J. If x P Q ℓ ˆ, we denote by Q ℓ rxs the 1-dimensional representation of the Weil group W where the inertia acts trivially and Frob acts like multiplication by the scalar x. Let τ 1 :" c ´Ind J N J pJ 1 q Ą ρ ∆ 2 where N J pJ 1 q is the normalizer of J 1 , and Ą ρ ∆ 2 is an extension to N J pJ 1 q of the cuspidal representation ρ ∆ 2 of J 1 . Then τ 1 is an irreducible supercuspidal representation of J. If Π P A ξ pIq, we define δ Πp :" ω Πp pp ´1 ¨idqp ´wpξq P Q ℓ ˆwhere ω Πp is the central character of Π p , and p ´1 ¨id lies in the center of J. For any isomorphism ι :

Q ℓ » C we have |ιpδ Πp q| " 1.
Theorem (3.5.2.3). There are GpA p f q ˆW -equivariant isomorphisms

H 0 c pSpb 0 q b F, L ξ q » à ΠPA ξ pIq ΠpPX un pJq Π p b Q ℓ rδ Πp p wpξq s, H 1 c pSpb 0 q b F, L ξ q » à ΠPA ξ pIq DχPX un pJq, Πp"χ¨St J Π p b Q ℓ rδ Πp p wpξq s ' à ΠPA ξ pIq DχPX un pJq, Πp"χ¨τ 1 Π p b Q ℓ r´δ Πp p wpξq`1 s, H 2 c pSpb 0 q b F, L ξ q » à ΠPA ξ pIq Π J 1 p "0 Π p b Q ℓ rδ Πp p wpξq`2 s.
Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

1.6 Adapting the approach to the ramified case

As one of the rapporteur wittingly pointed out to me, an unfortunate typo can be found in [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF], where the authors wrote in the course of a paragraph that Pappas' integral model of the unitary PEL GUp1, n ´1q Shimura variety, at a ramified prime and at parahoric level given by the stabilizer of a self-dual lattice, is smooth. This is however not the case, and it is consistent with such a parahoric subgroup being special but not hyperspecial, nor does it correspond to a case of exotic good reduction. We note that this typo has absolutely no impact on the contents of [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF], as smoothness is not needed there anyway.

In an earlier version of this thesis, I intended to apply the same approach as in the inert case to the ramified case, and with the smoothness hypothesis it seemed like all steps described above would work the same way. But, this hypothesis being actually wrong, the situation of the ramified case is more complex. Therefore, regarding the ramified case, this thesis only contains the part exploiting Deligne-Lusztig theory in order to get information on the cohomology of a closed Bruhat-Tits stratum.

Let us explain what obstacles we face and propose a slightly unformal strategy to overcome them.

The main issue concerns 3.4.1.5 Proposition, where one identifies the cohomology of an analytical tube U Λ with the cohomology of its special fiber M Λ . In the proof, we push the Bruhat-Tits stratum into the associated Shimura variety via p-adic uniformization, and we apply a result of Berkovich in [START_REF] Berkovich | Vanishing cycles for formal schemes. II[END_REF]. Since the integral model of the Shimura variety is smooth (in the inert case), nearby cycles are trivial and we obtain an isomorphism H ' pU Λ , Q ℓ q » H ' pM Λ , Q ℓ q. In general however, we only get an isomorphism with the cohomology of M Λ with coefficients in the nearby cycles sheaf RΨ η Q ℓ . In situations where these nearby cycles can not be computed, it seems hopeless to try using the approach of this thesis. Thus, in the ramified case, the non-smoothness of the integral model means that one must first understand the nearby cycles on the Shimura variety. The integral model S K p over SpecpO E q (with E{Q p quadratic ramified) has been built by Pappas in [START_REF] Pappas | On the arithmetic moduli schemes of PEL Shimura varieties[END_REF] as a moduli space classifying abelian schemes with usual additional structures, and satisfying a certain "Pappas condition" similar to the one in the definition of the Rapoport-Zink space in the ramified case. Theorem 4.5 of [START_REF] Pappas | On the arithmetic moduli schemes of PEL Shimura varieties[END_REF] states that S K p is normal, Cohen-Macaulay and flat over SpecpO E q. Moreover its special fiber S K p is smooth outside of a finite number of singular points, and if n ě 3 the blow-up BLpS K p q Ñ S K p at the reduced singular locus has semistable reduction, ie. it is regular and its special fiber is a divisor with normal crossings. If n " 2 then S K p already has semistable reduction. In this context, the nearby cycles R q Ψ η Q ℓ for q ą 0 are skyscraper and concentrated on the singular points of the special fiber S K p . For any point x P S K p , there exists a point y P M loc (the local model associated to the PEL datum) such that x and y have some isomorphic etale neighborhoods. The stalk pR q Ψ η Q ℓ q x may therefore be computed on the local model, which has a much simpler linear algebraic description. In fact, from general theory the integral model S K p of the Shimura variety and the local model M loc share the same geometric properties, and it is mainly the local model which is studied in [START_REF] Pappas | On the arithmetic moduli schemes of PEL Shimura varieties[END_REF] ; it is also described in [START_REF] Krämer | Local Models for Ramified Unitary Groups[END_REF]. Since the blow-up BlpM loc q Ñ M loc at the singular points has semistable reduction, we understand the nearby cycles on the blow-up (for instance via [START_REF] Illusie | Exposé I : Autour du théorème de monodromie locale[END_REF]), and via proper base change we could compute the cycles on the local model itself. Then, if one may understand the distribution of singular points of the special fiber S K p with respect to the Bruhat-Tits stratification on the supersingular locus, it seems reasonable to think that the cohomology groups H ' pM Λ , RΨ η Q ℓ q could be understood, at least sufficiently enough in order to apply the approach described in the inert case.

Organization of the thesis

The body of the thesis consists of the two papers written during the PhD, along with the first part of the 3rd paper. Section 2 is [START_REF] Muller | Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature (1,n-1)[END_REF], section 3 is [START_REF] Muller | Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature (1,n-1)[END_REF] and Section 4 is the first part of [START_REF] Muller | On the Cohomology of the Ramified PEL Unitary Rapoport-Zink Space of Signature[END_REF]. Section 2 and 3 deal with the inert case, the former consists of step (A) and the latter of step (B) as explained in the introduction. Section 4 deals with step (A) in the ramified case. Each section may be read independently, however some parts of them may make reference to previous sections as the papers have been written in this chronological order. We warn the reader that the notations may vary slightly from one section to the other, as well as they may vary from the introduction.

2 Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature p1, n ´1q

Notations

Throughout this section paper, we fix q a power of an odd prime number p. If k is a perfect field extension of F q , we denote by σ : x Þ Ñ x q the q-th power Frobenius of Galpk{F q q. We fix an algebraic closure F of F q . Unless specified otherwise, G will denote a connected reductive group over F equipped with an F q -structure, induced by a Frobenius morphism

F : G Ñ G.
If H is an F -stable subgroup of G, we denote by H :" H F » HpF q q its group of F q -rational points. We fix a pair pT, Bq consisting of a maximal torus T contained in a Borel subgroup B, both of them being F -stable. Such a pair always exists up to G " G F -conjugation. We obtain a Coxeter system pW, Sq on which F acts, where W " WpTq is the Weyl group attached to T and S is the set of simple reflexions. It can be identified with the Weyl group of G as defined in [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF]. Let ℓ denote the length function on W relative to S. For I Ă S, we write P I , U I , L I respectively for the standard parabolic subgroup of type I, for its unipotent radical and for its unique Levi complement containing T. We also write W I for the parabolic subgroup of W generated by the simple reflexions in I. Recall that an element w P W is said to be I-reduced (resp. reduced-I) if for every v P W I , we have ℓpvwq " ℓpvq `ℓpwq (resp. ℓpwvq " ℓpwq `ℓpvq).

The set of I-reduced (resp. reduced-I) elements is denoted by I W (resp. W I ). If I, I 1 Ă S, an element is said to be I-reduced-I 1 if it belongs to I W I 1 :" I W X W I 1 .

2.1

The generalized Deligne-Lusztig variety X I pidq 2.1.1 Let G be a connected reductive group over F. Let F be a Frobenius morphism defining an F q -structure on it. If H is an F -stable subgroup of G, we denote by H :" H F » HpF q q its group of F q -rational points. We fix a pair pT, Bq consisting of a maximal torus T contained in a Borel subgroup B, both of them being F -stable. Such a pair always exists up to G " G Fconjugation. We obtain a Coxeter system pW, Sq on which F acts, where W " WpTq is the Weyl group attached to T and S is the set of simple reflexions. It can be identified with the Weyl group of G as defined in [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF]. Let ℓ denote the length function on W relative to S. For I Ă S, we write P I , U I , L I respectively for the standard parabolic subgroup of type I, for its unipotent radical and for its unique Levi complement containing T. We also write W I for the parabolic subgroup of W generated by the simple reflexions in I. Recall that an element w P W is said to be I-reduced (resp. reduced-I) if for every v P W I , we have ℓpvwq " ℓpvq`ℓpwq (resp. ℓpwvq " ℓpwq `ℓpvq). The set of I-reduced (resp. reduced-I) elements is denoted by I W (resp. W I ). If I, I 1 Ă S, an element is said to be I-reduced-I 1 if it belongs to I W I 1 :" I W X W I 1 .

2.1.2

We recall the definition of Deligne-Lusztig varieties from [START_REF] Bonnafé | On the irreducibility of Deligne-Lusztig varieties[END_REF]. If P is any parabolic subgroup of G, the associated generalized parabolic Deligne-Lusztig variety is

X P :" tgP P G{P | g ´1F pgq P PF pPqu.
When these varieties were first introduced in [DL76] only the case of Borel subgroups was considered, hence the adjective "parabolic". Moreover, parabolic Deligne-Lusztig varieties have mostly been studied with the additional assumption that P contains an F -stable Levi complement, see for instance [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF]. This is not required by the definition above, hence the adjective "generalized". Using the Coxeter system as above, one may give an equivalent description of these varieties. For I, I 1 Ă S the generalized Bruhat decomposition is an isomorphism

P I zG{P I 1 " ğ wP I W I 1 P I zP I wP I 1 {P I 1 » W I zW{W I 1 .
For w P I W F pIq , the generalized parabolic Deligne-Lusztig varieties is defined by X I pwq " tgP I P G{P I | g ´1F pgq P P I wF pP I qu.

The families of varieties X P and X I pwq are the same and [START_REF] Bonnafé | On the irreducibility of Deligne-Lusztig varieties[END_REF] explains how to go from one description to the other. The case I " H corresponds to usual Deligne-Lusztig varieties in G{B.

Moreover, the additional assumption regarding the existence of a rational Levi complement translates into the equation

w ´1I w " F pIq, (˚)
which is a compatibility condition between the parameters w and I. The variety X I pwq is defined over F q ι , where ι is the least integer such that F ι pIq " I and F ι pwq " w.

2.1.3

In this paragraph, we compute the dimension of a generalized Deligne-Lusztig variety X I pwq. For any w P W, let ℓpwq denote the length of w with respect to S.

Proposition. For I Ă S and w P I W F pIq , we have

dim X I pwq " ℓpwq `dim G{P IXwF pIqw ´1 ´dim G{P I .
Let us introduce a few more notations. If I, I 1 Ă S, the generalized Bruhat decomposition implies that the G-orbits for the diagonal action on G{P I ˆG{P I 1 are given by O I,I 1 pwq :" tpgP I , hP I 1 q | g ´1h P P I wP I 1 u for w P I W I 1 . The Deligne-Lusztig variety X I pwq can be seen as the intersection of O I,F pIq pwq with the graph of the Frobenius F : G{P I Ñ G{P F pIq . This intersection is transverse, see [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF] 9.11 (in loc. cit. the proof deals with the case I " H, but it generalizes to any I). Thus, the proposition follows from the following lemma and the fact that dim P I " dim P F pIq .

Lemma. For I, I 1 Ă S and w P I W I 1 , we have

dim O I,I 1 pwq " ℓpwq `dim G{P IXwI 1 w ´1 .
Proof. Recall that for I Ă S, the standard parabolic subgroup of type I decomposes as a union of Bruhat cells P I " BW I B, and any Bruhat cell BwB has dimension dim B`ℓpwq. Therefore dim P I " dim B `ℓpIq where ℓpIq denotes the maximal length of elements of W I . Let I, I 1 and w be as in the lemma. Consider the first projection O I,I 1 pwq Ñ G{P I which is a surjective morphism with fibers isomorphic to P I wP I 1 {P I 1 . It is flat since G Ñ G{P I is faithfully flat, and the pullback O I,I 1 pwq ˆG{P I G is isomorphic to G ˆPI wP I 1 {P I 1 . We have

P I wP I 1 " BW I BwBW I 1 B " BW I wW I 1 B,
therefore the dimension of a fiber is given by dim P I wP I 1 {P I 1 " dim P I wP I 1 ´dim P I 1 " max vPW I wW I 1 ℓpvq ´ℓpI 1 q.

Since w is I-reduced-I 1 , according to [DM20] Lemma 3.2.2, any element v P W I wW I 1 can uniquely be written as v " xwy such that x P W I , y P W I 1 and xw is reduced-I 1 . In particular ℓpvq " ℓpxq `ℓpwq `ℓpyq. It follows that max

vPW I wW I 1
ℓpvq " ℓpwq `max xPW I XW I 1 w ´1 ℓpxq `ℓpI 1 q.

We prove that W I X W I 1 w ´1 " W I X W IXwI 1 w ´1 .

Let x P W I X W I 1 w ´1, we show that x is reduced-I X wI 1 w ´1. Let s P I X wI 1 w ´1, so that we can write s " wtw ´1 for some t P I 1 . Then xsw " xwt. Since xs P W I and w is I-reduced, the left hand side has length ℓpxsq `ℓpwq. On the other hand, since t P I 1 and xw is reduced-I 1 , the right hand side has length ℓpxwq `1 " ℓpxq `ℓpwq `1. Therefore ℓpxsq " ℓpxq `1 as expected.

For the other inclusion, let y P W I X W IXwI 1 w ´1 . We show that yw is reduced-I 1 . Towards a contradiction, assume that ℓpywtq ă ℓpywq for some t P I 1 . Let y " s 1 . . . s r and w " u 1 . . . u r 1 be reduced expressions respectively of y and of w, with the s i in I and the u j in S. Since w is I-reduced, the concatenation of both reduced expressions give a reduced expression of yw. By the exchange condition (see [START_REF] Digne | Representations of Finite Groups of Lie Type[END_REF] 2.1.2), we have ywt " s 1 . . . p s i . . . s r w or yu 1 . . . p u j . . . u r 1

for some 1 ď i ď r or 1 ď j ď r 1 , where p ¨denotes the product with one omitted term.

The second case is impossible, since after simplifying y it would contradict the fact that w is reduced-I 1 . Let us write s :" y ´1s 1 . . . p s i . . . s r P W I , so that we have wt " sw.

The left hand side has length ℓpwq `1, and the right hand side has length ℓpsq `ℓpwq. It follows that s P I has length 1. Therefore s " wtw ´1 P I X wI 1 w ´1. Eventually, we have ℓpysq " ℓpyq `1 since y is reduced-pI X wI 1 w ´1q. This is absurd, because ys " s 1 . . . p s i . . . s r has length r ´1 " ℓpyq ´1.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

To conclude the proof, we recall the following general fact. If pW, Sq is a Coxeter system and K Ă S, then the product map W K ˆWK " Ý Ñ W mapping pw K , w K q to w K w K is a bijection. In particular we have max wPW ℓpwq " max

w K PW K ℓpw K q `max w K PW K ℓpw K q.
We apply this to the Coxeter system pW I , Iq and K " I X wI 1 w ´1. It follows that max xPW I XW I 1 w ´1 ℓpxq " max xPW I XW IXwI 1 w ´1 ℓpxq " ℓpIq ´ℓpI X wI 1 w ´1q.

Putting things together, have proved that dim O I,I 1 pwq " dim G{P I `dim P I wP

I 1 {P I 1 " dim G ´dim B ´ℓpIq `max vPW I wW I 1 ℓpvq ´ℓpI 1 q " dim G ´dim B ´ℓpIq `ℓpwq `max xPW I XW I 1 w ´1 ℓpxq " dim G ´dim B ´ℓpI X wI 1 w ´1q `ℓpwq " dim G{P IXwI 1 w ´1 `ℓpwq.
Remark. In [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 4.4, the formula given by the authors for the dimension of O I,I 1 pwq, and as a consequence for the Deligne-Lusztig variety X I pwq as well, contained a mistake.

2.1.4

Let d be a nonnegative integer and let V be a p2d `1q-dimensional F q 2 -vector space. Let p¨, ¨q : V ˆV Ñ F q 2 be a non-degenerate hermitian form on V . This hermitian structure on V is unique up to isomorphism. In particular, we may once and for all a basis B of V in which p¨, ¨q is described by the square matrix 9 w 0 of size 2d `1, having 1 on the anti-diagonal and 0 everywhere else. If k is a perfect field extension of F q 2 , we may extend the pairing to V k :" V b F q 2 k by setting pv b x, w b yq :" xy σ pv, wq P k for all v, w P V and x, y P k. If U is a subspace of V k we denote by U K its orthogonal, that is the subspace of all vectors x P V k such that px, U q " 0. Let J denote the finite group of Lie type UpV, p¨, ¨qq. It is defined as the group of F -fixed points of J :" GLpV q F with F a non-split Frobenius morphism. Using the basis B, the group J is identified with GL 2d`1 with F q -structure induced by the Frobenius morphism F pM q :" 9 w 0 pM pqq q ´t 9 w 0 . Here, M pqq denotes the matrix M having all coefficients raised to the power q. We may then identify J with the usual finite unitary group U 2d`1 pqq. The pair pT, Bq consisting of the maximal torus of diagonal matrices and the Borel subgroup of upper-triangular matrices is F -stable. The Weyl system of pT, Bq may be identified with pS 2d`1 , Sq in the usual manner, where S is the set of simple transpositions s i :" pi i `1q for 1 ď i ď 2d. Under this identification, the Frobenius acts on W as the conjugation by the element w 0 , characterized for having the maximal length. It satisfies w 0 piq " 2d `2 ´i, and a natural representative of w 0 in the normalizer of T is no other than 9 w 0 . Since w 0 has order 2, the action of the Frobenius on W is involutive. It also preserves the simple reflexions with the formula F ps i q " s 2d`1´i .

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q 2.1.5 We define the following subset of S I :" ts 1 , . . . , s d , s d`2 , . . . , s 2d u " Szts d`1 u.

We have F pIq " Szts d u " I. We consider the generalized Deligne-Lusztig variety X I pidq. It corresponds to the variety denoted Y Λ in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 4.5. It has dimension d and it does not satisfy the compatibility condition (˚).

Proposition ([VW11] 4.4). The variety X I pidq is defined over F q 2 and it is projective, smooth, geometrically irreducible of dimension d.

Although the proposition in loc. cit. is only stated in the case q " p, the arguments carry over to general q. The geometric irreducibility is a consequence of the criterion proved in [START_REF] Bonnafé | On the irreducibility of Deligne-Lusztig varieties[END_REF].

Remark. Even though the dimension formula for generalized Deligne-Lusztig varieties in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] is wrong, it does give the correct result in the case of X I pidq. It is because for w " id, we have I X wF pIqw ´1 " I X F pIq. Therefore, that mistake does not change anything regarding the validity of the authors' work. For example, we may consider the Deligne-Lusztig variety X I ps 2 s 1 q for U 3 pF q q with I " ts 1 u. It is classical so that dim X I ps 2 s 1 q " ℓps 2 s 1 q " 2. However, we have P IXF pIq " B and dim G{B " 3 whereas dim G{P I " 2, so that the formula of loc. cit. says that X I ps 2 s 1 q would be of dimension 2 `3 ´2 " 3.

2.1.6

Rational points of Deligne-Lusztig varieties associated to a unitary group U over F q can be described in terms of vectorial flags, in a certain relative position with respect to their image by the Frobenius. Let k be a perfect field extension of F q 2 . According to [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] 2.12, the Frobenius acts on a flag F in V k by sending it to its orthogonal flag F K . Explicitely, we have

F : t0u Ă F 1 Ă . . . Ă F r Ă V k , F K : t0u Ă F K r Ă . . . Ă F K 1 Ă V k .
Here, given our choice of I, a k-rational point of X I pidq corresponds to a flag of the type

F : t0u Ă U Ă V k
with U having dimension d `1, and which is of relative position id with respect to F K . This precisely means that U must contain U K .

Proposition. The k-rational points of X I pidq are given by

X I pidqpkq » tU Ă V k | dim U " d `1 and U K Ă U u.
2.1.7 In [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 5.3, the authors defined the Ekedahl-Oort stratification on the Deligne-Lusztig variety X I pidq. By loc. cit. Corollary 5.12, it turns out that each stratum is itself isomorphic to a parabolic Deligne-Lusztig variety which is not generalized. They are defined Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q as follows. For 0 ď t ď d, we define the subset I t :" ts 1 , . . . , s d´t´1 , s d`t`2 , . . . , s 2d u Ă S.

The subset I t consists of all 2d simple reflexions in S, except that we removed the 2t `2 ones in the middle. Thus, it has cardinality 2pd ´t ´1q. In particular, it is empty for t " d or d ´1. We also define the cycle w t :" pd `t `1 d `t . . . d `1q. Its decomposition into simple reflexions is w t " s d`1 . . . s d`t . When t " 0, it is the identity. We note that even though I d " I d´1 " H, we still have w d " w d´1 . One may check that F pI t q " I t and that w t belongs to It W It . Moreover, the compatibility condition (˚) is satisfied for the pair pI t , w t q. Indeed, the reduced decomposition for w t does not use any simple reflexion that is adjacent to those in I t .

Proposition ([VW11] 3.3 and 5.3). The Deligne-Lusztig variety X It pw t q is defined over F q 2 and has dimension t. There is a natural immersion X It pw t q ãÑ X I pidq inducing a stratification

X I pidq " ğ 0ďtďd X It pw t q.
The closure of the stratum X It pw t q is the union of all the strata X Is pw s q for s ď t.

2.1.8 Following the proof of Theorem 2.15 of [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF], we can describe the stratification at the level of rational points. Let k be a perfect field extension of F q 2 . Because of the choice of I t , a k-point of X It pw t q is a flag

F : t0u Ă F ´t´1 Ă . . . Ă F ´1 Ă F 1 Ă . . . Ă F t`1 Ă V k
with dimpF ´iq " d `1 ´i and dimpF i q " d `i for 1 ď i ď t `1, and which is in relative position w t with respect to F K . It means that we have a diagram of the following type.

F : F ´t´1 Ă . . . Ă F ´1 Ă F 1 Ă F 2 Ă . . . Ă F t Ă F t`1 F K : F K t`1 Ă . . . Ă F K 1 Ă τ pF 1 q Ă τ pF 2 q Ă . . . Ă τ pF t q Ă τ pF t`1 q { { {
Here, τ :" σ 2 ¨id is an F q 2 -linear automorphism of V k , and it satisfies τ pU q " pU K q K for every subspace U Ă pV Λ q k . This diagram implies that τ pF i q " F i´1 `τ pF i´1 q for all 2 ď i ď t `1. This rewrites as F i " F i´1 `τ ´1pF i´1 q. We deduce that

F i " i´1 ÿ l"0 τ ´lpF 1 q
for all 1 ď i ď t `1. Thus, the whole flag is determined by the subspace F 1 , which has dimension d `1 and contains its orthogonal. The immersion X It pw t q ãÑ X I pidq maps the flag F to F 1 .

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

Conversely, a k-point of X I pidq is given by a subspace U Ă V k of dimension d `1 containing its orthogonal. For i ě 1 we define

F i :" i´1 ÿ l"0 τ ´1pU q Ă V k .
Then pF i q iě1 is a nondecreasing sequence of subspaces of V k . Let t be the smallest integer such that F t`1 " F t`2 . It follows that 0 ď t ď d and that t is also the smallest integer such that F t`1 " τ pF t`1 q. Moreover the orthogonal U K has dimension d and we have U K Ă U , so that U K Ă pU K q K " τ pU q. In particular, if t ą 0 then U X τ pU q " U K . Thus, we have dimpF 2 q " d `2. Similarly, we have dimpF i q " d `i for all 1 ď i ď t `1. By setting F ´i :" F K i , we obtain a flag F that is the k-rational point of X It pw t q associated to U .

2.1.9 The Deligne-Lusztig varieties X It pw t q are related to Coxeter varieties for smaller unitary groups as we now explain. We define K t :" ts 1 , . . . s d´t´1 , s d´t`1 , . . . , s d`t , s d`t`2 , . . . , s 2d u " Szts d´t , s d`t`1 u.

The set K t is obtained from I t by adding the 2t simple reflexions in the middle. It has cardinality 2d ´2 and satisfies F pK t q " K t . We have I t Ă K t with equality if and only if t " 0.

Proposition. There is a U 2d`1 pqq-equivariant isomorphism

X It pw t q » U 2d`1 pqq{U Kt ˆLK t X L K t It pw t q,
where X

L K t
It pw t q is a Deligne-Lusztig variety for L Kt . The zero-dimensional variety U 2d`1 pqq{U Kt has a left action of U 2d`1 pqq and a right action of L Kt .

Proof. This is an application of [DM14] Proposition 7.19 which is the geometric identity behind the transitivity of the Deligne-Lusztig functors. It applies to the varieties X It pw t q because they satisfy the compatibility condition (˚), and satisfies the following conditions: K t contains I t , it is stable by the Frobenius and w t belongs to the parabolic subgroup W Kt » S d´t ˆS2t`1 ˆSd´t Ă S 2d`1 .

2.1.10

The Levi complement L Kt is isomorphic to the product GL d´t ˆGL 2t`1 ˆGL d´t as a reductive group over F. Given a matrix M " diagpA, C, Bq P L Kt , we have F pM q " diagpF pBq, F pCq, F pAqq, where we still denote by F the Frobenius morphism for smaller linear groups. Writing H for the product of the two GL d´t factors, we have L Kt » H ˆGL 2t`1 and both factors inherit an F q -structure by means of F . We have L Kt » GL d´t pq 2 q ˆU2t`1 pqq, the first factor corresponding to H. The Weyl group of L Kt is isomorphic to W H ˆS2t`1 where W H » S d´t ˆSd´t is the Weyl group of H. Via this decomposition, the permutation w t corresponds to id ˆr w t , where r w t is the restriction of w t to td ´t `1, . . . , d `t `1u. Similarly, the set of simple reflexions S decomposes as S H \ r S, the second term corresponding to the simple reflexions in S 2t`1 . Then, we have

I t " S H \ H.
The Deligne-Lusztig variety for L Kt decompose accordingly as the following product

X L K t It pw t q " X H S H pidq ˆXU 2t`1 pqq H p r w t q.
The variety X H S H pidq is just a point, whereas X U 2t`1 pqq H p r w t q is a Deligne-Lusztig variety for the unitary group of size 2t`1. We observe that the permutation r w t is a Coxeter element in S 2t`1 , ie. the product of exactly one simple reflexion for each orbit of the Frobenius. Deligne-Lusztig varieties attached to Coxeter elements are called Coxeter varieties, and their cohomology with coefficients in Q ℓ where ℓ is a prime number different from p are well understood thanks to the work of Lusztig in [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF]. Before stating the results of loc. cit. we recall parts of the representation theory of finite unitary groups.

2.2 Irreducible unipotent representations of the finite unitary group 2.2.1 In this section, we recall the classification of the irreducible unipotent representations of the finite unitary group and we explain the underlying combinatorics. We use the notations from 2.1.1. For w P W, let 9 w be a representative of w in the normalizer N G pTq of T. By the Lang-Steinberg theorem, one can find g P G such that 9

w " g ´1F pgq. Then g T :" gTg ´1 is another F -stable maximal torus, and w P W is said to be the type of g T with respect to T. Every F -stable maximal torus arises in this manner. According to [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF] Corollary 1.14, the G-conjugacy class of g T only depends on the F -conjugacy class of the image w of the element g ´1F pgq P N G pTq in the Weyl group W. Here, two elements w and w 1 in W are said to be F -conjugates if there exists some element u P W such that w " uw 1 F puq ´1. For every w P W, we fix T w an F -stable maximal torus of type w with respect to T. The Deligne-Lusztig induction of the trivial representation of T w is the virtual representation of G defined by the formula R w :"

ÿ iě0 p´1q i H i c pX H pwqq
where X H pwq is a Deligne-Lusztig variety for G as defined in 2.1.2. According to [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF] Theorem 1.6, the virtual representation R w only depends on the F -conjugacy class of w in W.

An irreducible representation of G is said to be unipotent if it occurs in R w for some w P W.

The set of isomorphism classes of unipotent representations of G is usually denoted EpG, 1q following Lusztig's notations.

2.2.2

Assume that the Coxeter graph of the reductive group G is a union of subgraphs of type A m (for various m). Let | W be the set of isomorphism classes of irreducible representations of its Weyl group W. The action of the Frobenius F on W induces an action on | W, and we consider the fixed point set | W F . Then, the following classification theorem is well known.

Theorem ([LS77] Theorem 2.2).

There is a bijection between | W F and the set of isomorphism classes of irreducible unipotent representations of G " G F .

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

We recall how the bijection is constructed. If V P | W F is an irreducible F -stable representation of W, according to loc. cit. there is a unique automorphism r F of V of finite order such that

RpV q :" 1 |W| ÿ wPW Tracepw ˝r F | V qR w
is an irreducible representation of G. Then the map V Þ Ñ RpV q is the desired bijection. In the case G " GL n with the Frobenius morphism F being either standard or twisted (ie. G " GL n pqq or U n pqq), we have an equality | W F " | W. Moreover, the automorphism r F is the identity in the former case and multiplication by w 0 on the latter, where w 0 is the element of maximal length in W. Thus, in both cases the irreducible unipotent representations of G are classified by the irreducible representations of the Weyl group W » S n , which in turn are classified by partitions of n or equivalently by Young diagrams. We now recall the underlying combinatorics behind the representation theory of the symmetric group. A general reference is [START_REF] James | The Representation Theory of the Symmetric Group[END_REF].

2.2.3

A partition of n is a tuple λ " pλ 1 ě . . . ě λ r q with r ě 1 and the λ i 's are positive integers such that λ 1 `. . . `λr " n. The integer n is called the length of the partition and it is also denoted by |λ|. If a partition has a series of repeating integers, it is common to write it shortly with an exponent. For instance, the partition p3, 3, 2, 2, 1q of 11 will be denoted p3 2 , 2 2 , 1q. Partitions of n are naturally identified with Young diagrams of size n. The diagram attached to λ has r rows consisting successively of λ 1 , . . . , λ r boxes. To any partition λ of n, one can naturally associate an irreducible representation χ λ of the symmetric group S n . An explicit construction is given, for instance, by the notion of Specht modules as explained in [Jam84] 7.1. In particular, the character χ pnq is trivial while the character χ p1 n q is the signature.

2.2.4

We recall the Murnaghan-Nakayama rule which gives a recursive formula to evaluate the characters χ λ . We first need to introduce skew Young diagrams. Consider a pair λ and µ of two partitions respectively of integers n `k and k. Assume that the Young diagram of µ is contained in the Young diagram of λ. By removing the boxes corresponding to µ from the diagram of λ, one finds a shape consisting of n boxes denoted by λzµ. Any such shape is called a skew Young diagram of size n. It is said to be connected if one can go from a given box to any other by moving in a succession of adjacent boxes. For example, consider the partition λ " p3 2 , 2 2 , 1q and let us define the partitions µ 1 " p2 2 q, µ 2 " p3, 1 2 q and µ 3 " p2, 1q. The diagrams below correspond, from left to right, to the skew Young diagrams λzµ i for i " 1, 2, 3.

The skew Young diagram λzµ 1 is not connected, whereas the others are connected. A skew Young diagram is said to be a border strip if it is connected and if it does not contain any 2 ˆ2 square. The height of a border strip is defined as its number of rows minus 1. For instance, among the three skew Young diagrams above only λzµ 2 is a border strip. Its size is 6 and its height is 3. The characters χ λ are class functions, so we only need to specify their values on conjugacy classes of the symmetric group S n . These conjugacy classes are also naturally labelled by partitions of n. Indeed, up to ordering any permutation σ P S n can be uniquely decomposed as a product of r ě 1 cycles c 1 , . . . , c r with disjoint supports. We denote by ν i the cycle length of c i and we order them so that ν 1 ě . . . ě ν r . We allow cycles to have length 1, so that the union of the supports of all the c i 's is t1, . . . , nu. Thus, we obtain a partition ν " pν 1 , . . . , ν r q of n which is called the cycle type of the permutation σ. Two permutations are conjugates in S n if and only if they share the same cycle type. We denote by χ λ pνq the value of the character χ λ on the conjugacy class labelled by ν.

Theorem (Murnaghan-Nakayama rule). Let λ and ν be two partitions of n. We have

χ λ pνq " ÿ S p´1q htpSq χ λzS pνzν 1 q,
where S runs over the set of all border strips of size ν 1 in the Young diagram of λ, such that removing S from λ gives again a Young diagram. Here, the integer htpSq P Z ě0 is the height of the border stip S, the Young diagram λzS is the one obtained by removing S from λ, and νzν 1 is the partition of n ´ν1 obtained by removing ν 1 from ν.

Applying the Murnaghan-Nakayama rule in successions results in the value of χ λ pνq. We see in particular that χ pnq is the trivial character whereas χ p1 n q is the signature. We illustrate the computations with λ " p3 2 , 2 2 , 1q and ν " p4 2 , 3q. There are only two elligible border strips of size 4 in the diagram of λ, as marked below.

ˆand ˆB oth border strips have height 2. Thus, the formula gives χ p3 2 ,2 2 ,1q p4 2 , 3q " χ p3 2 ,1q p4, 3q `χp3,1 4 q p4, 3q.

In each of the two Young diagrams obtained after removal of the border strips, there is only one elligible strip of size 4, and eventually the three last remaining boxes form the final border strip of size 3.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q ˆˆˆù ñ ˆˆˆˆù ñ ˆˆT aking the heights of the border strips into account, we find χ p3 2 ,1q p4, 3q " ´χp3q p3q " ´χH " ´1, χ p3,1 4 q p4, 3q " ´χp3q p3q " ´χH " ´1.

Here, H denotes the empty partition. The computation finally gives χ p3 2 ,2 2 ,1q p4 2 , 3q " ´2.

2.2.5

The irreducible unipotent representation of U n pqq (resp. GL n pqq) associated to χ λ by the bijection of 2.2.1 Theorem is denoted by ρ U λ (resp. ρ GL λ ). The partition pnq corresponds to the trivial representation and p1 n q to the Steinberg representation in both cases. We will omit the superscript when the group we are talking about is clear from the context. The degrees of the representations ρ GL λ and ρ U λ are given by expressions known as hook formula. Given a box l in the Young diagram of λ, its hook length hp l q is 1 plus the number of boxes lying below it or on its right. For instance, in the following figure the hook length of every box of the Young diagram of λ " p3 2 , 2 2 , 1q has been written inside it. Proposition ([GP00] Propositions 4.3.1 and 4.3.5). Let λ " pλ 1 ě . . . ě λ r q be a partition of n. The degrees of the irreducible unipotent representations ρ GL λ and ρ U λ , respectively of GL n pqq and U n pqq, are given by the following formulas

degpρ GL λ q " q apλq ś n i"1 q i ´1 ś lPλ q hplq ´1 , degpρ U λ q " q apλq ś n i"1 q i ´p´1q i ś lPλ q hplq ´p´1q hplq ,
where apλq " ř r i"1 pi ´1qλ i .

2.2.6

We recall from [GM20] 3.1 and 3.2 some definitions on classical Harish-Chandra theory.

A parabolic subgroup of G is a subgroup P Ă G such that there exists an F -stable parabolic subgroup P of G with P " P F . A Levi complement of G is a subgroup L Ă G such that there exists an F -stable Levi complement L of G, contained inside some F -stable parabolic subgroup, such that L " L F . Any parabolic subgroup P of G has a Levi complement L.

Let L " L F be a Levi complement of G inside a parabolic subgroup P " P F . Let U " U F be the F -fixed points of the unipotent radical U of P. The Harish-Chandra induction and restriction functors are defined by the following formulas.

R G LĂP : ReppLq Ñ ReppGq ˚RG LĂP : ReppGq Ñ ReppLq σ Þ Ñ CrG{U s b CrLs σ ρ Þ Ñ Hom G pCrG{U s, ρq
Here, ReppGq is the category of complex representations of G, and similarily for ReppLq. These two functors are adjoint, and up to isomorphism they do not depend on the choice of the parabolic subgroup P containing the Levi complement L. For this reason, we will denote the functors R G L and ˚RG L instead. An irreducible representation of G is called cuspidal if its Harish-Chandra restriction to any proper Levi complement is zero. We consider pairs pL, Xq where L is a Levi complement of G and X is an irreducible representation of L. We define an order on the set of such pairs by setting pL, Xq ď pM, Y q if L Ă M and if X occurs in the Harish-Chandra restriction of Y to L. A pair is said to be cuspidal if it is minimal with respect to this order, in which case X is a cuspidal representation of L. If pL, Xq is a cuspidal pair, we will denote by rL, Xs its conjugacy class under G. Given a cuspidal pair pL, Xq of G, its associated Harish-Chandra series EpG, pL, Xqq is defined as the set of isomorphism classes of irreducible constituents in the induction of X to G. Each series is non empty. Two of them are either disjoint or equal, the latter occuring if and only if the two cuspidal pairs are conjugates in G. Thus, the series are indexed by the conjugacy classes of cuspidal pairs rL, Xs. Moreover, the isomorphism class of any irreducible representation of G belongs to some Harish-Chandra series. Thus, Harish-Chandra series form a partition of the set of isomorphism classes of irreducible representations of G. If ρ is an irreducible representation of G, the conjugacy class rL, Xs corresponding to the series to which ρ belongs is called the cuspidal support of ρ. If T denotes a maximal torus in G, then the series EpG, pT, 1qq is called the unipotent principal series of G.

2.2.7

For the general linear group GL n pqq, there is no unipotent cuspidal representation unless n " 1, in which case the trivial representation is cuspidal. Moreover, the unipotent representations all belong to the principal series. The situation for the unitary group is very different. First, by [START_REF] Lusztig | Irreducible representations of finite classical groups[END_REF] 9.2 and 9.4 there exists an irreducible unipotent cuspidal representation of U n pqq if and only if n is an integer of the form n " xpx`1q 2 for some x ě 0, and when that is the case it is the one associated to the partition ∆ x :" px, x ´1, . . . , 1q, whose Young diagram has the distinctive shape of a reversed staircase. Here, as a convention U 0 pqq denotes the trivial group. For example, here are the Young diagrams of ∆ 1 , ∆ 2 and ∆ 3 . Of course, the one of ∆ 0 the empty diagram.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q Furthermore the unipotent representations decompose non trivially into various Harish-Chandra series, as we recall from [GM20] 4.3. We consider an integer x ě 0 such that n decomposes as n " 2a `xpx`1q 2 for some a ě 0. We also consider the standard Levi complement L x » GL 1 pq 2 q a ˆU xpx`1q 2 pqq which corresponds to the choice of simple reflexions s a`1 , . . . , s n´a´1 . We write ρ x for the inflation of ρ U ∆x to an irreducible representation of L x . Then EpU n pqq, 1q decomposes as the disjoint union of all the Harish-Chandra series EpU n pqq, pL x , ρ x qq for all possible choices of x. With these notations, the principal unipotent series corresponds to x " 0 if n is even and to x " 1 if n is odd.

2.2.8

Given an irreducible unipotent representation ρ λ of U n pqq, there is a combinatorical way of determining the Harish-Chandra series to which it belongs. We consider the Young diagram of λ. We call domino any pair of adjacent boxes in the diagram. It may be either vertical or horizontal. We remove dominoes from the rim of the diagram of λ so that the resulting shape is again a Young diagram, until one can not proceed further. This process results in the Young diagram of the partition ∆ x for some x ě 0, and it is called the 2-core of λ. It does not depend on the successive choices for the dominoes. Then, the representation ρ λ belongs to the series EpU n pqq, pL x , ρ x qq if and only if λ has 2-core ∆ x . For instance, the diagram λ " p3 2 , 2 2 , 1q has 2-core ∆ 1 , as it can be determined by the following steps. We put crosses inside the successive dominoes that we remove from the diagram. Thus, the unipotent representation ρ λ of U 11 pqq belongs to the unipotent principal series EpU 11 pqq, pL 1 , ρ 1 qq. Theorem 3.2.5 of [START_REF] Geck | The Character Theory of Finite Groups of Lie Type: A Guided Tour[END_REF] establishes an isomorphism between the endomorphism algebra of the induced representation R G L pXq and the complex group ring of the ramification group W G pL, Xq.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

In particular, this gives an bijection between the Harish-Chandra series EpG, pL, Xqq and the set IrrpW G pL, Xqq of isomorphism classes of irreducible complex characters of W G pL, Xq. These bijections for G and for various Levi complements in G can be chosen to be compatible with Harish-Chandra induction. This is known as Howlett and Lehrer's comparison theorem which was proved in [START_REF] Howlett | Representations of Generic Algebras and Finite Groups of Lie Type[END_REF].

Theorem ([GM20] Comparison Theorem 3.2.7). Let pL, Xq be a cuspidal pair for the finite group of Lie type G. For every Levi complement M in G containing L, the bijection between IrrpW M pL, Xqq and EpM, pL, Xqq can be taken so that the diagrams

ZEpG, pL, Xqq ZIrrpW G pL, Xqq ZEpM, pL, Xqq ZIrrpW M pL, Xqq " " R G M Ind ZEpG, pL, Xqq ZIrrpW G pL, Xqq ZEpM, pL, Xqq ZIrrpW M pL, Xqq " ˚RG M
Res " are commutative. Here, Ind and Res on the right-hand side of the diagrams are the classical induction and restriction functors for representations of finite groups.

In other words, computing Harish-Chandra induction and restrictions of representations in G can be entirely done at the level of the associated Coxeter groups. In order to use this statement for unitary groups, we need to make the horizontal arrows explicit and to understand the combinatorics behind induction and restriction of the irreducible representations of the relevant Coxeter groups. This has been explained consistently in [START_REF] Fong | Brauer trees in classical groups[END_REF] for classical groups.

2.3.2

We focus on the case of the unitary group. Let x ě 0 such that n " 2a `xpx`1q 2 for some a ě 0. We consider the cuspidal pair pL x , ρ x q as in 2.2.7, with L x " GL 1 pq 2 q a ˆUxpx`1q 2 pqq. The relative Weyl group W Unpqq pL x q is isomorphic to the Coxeter group of type B a , which is usually denoted by W a . Indeed, the Weyl group W Unpqq pL x q admits a presentation by elements σ 1 , . . . , σ a´1 and θ of order 2 satisfying the relations

θσ 1 θσ 1 " σ 1 θσ 1 θ, θσ i " σ i θ, @ 2 ď i ď m ´1. σ i σ i`1 σ i " σ i`1 σ i σ i`1 , σ i σ j " σ j σ i , @ |i ´j| ě 2.
Explicitely, the element σ i is represented by the permutation matrix of the double transposition pi i `1qpn ´i n ´i `1q and the element θ by the matrix of the transposition p1 nq, all of which belong to N Unpqq pL x q. This presentation coincide with the Coxeter group W a of type B a , see in [GP00] 1.4.1. Moreover, the ramification group W Unpqq pL x , ρ x q is equal to the whole of W Unpqq pL x q » W a . The identification between the ramification group and the Coxeter group W a is naturally induced by the isomorphism between the absolute Weyl group W and the symmetric group S n . In order to proceed further, we need to explain the representation theory of the group W a .

2.3.3

Let W a be a Coxeter group of type B a given with a presentation by elements σ 1 , . . . , σ a´1 and θ satisfying equations as in 2.3.2. For 1 ď i ď a ´1, we define θ i " σ i . . . σ 1 θσ 1 . . . σ i . In particular θ 0 " θ. Following [GP00] 3.4.2, we define signed blocks to be elements of the following form. Given k ě 0 and e ě 1 such that k `e ď a, the positive (resp. negative) block of length e starting at k is

b k,e :" σ k`1 σ k`2 . . . σ k`e´1 , b ḱ,e :" θ k σ k`1 σ k`2 . . . σ k`e´1 .
A bipartition of a is an ordered pair pα, βq where α is a partition of some integer 0 ď j ď a and β is a partition of a ´j. Given a bipartition pα, βq of a and writing α " pα 1 , . . . , α r q and β " pβ 1 , . . . , β s q, we define the element

w α,β :" b ḱ1 ,β 1 . . . b ḱs,βs b ks`1 ,α 1 . . . b ks`r ,αr
where

k 1 " 0, k i`1 " k i `βi if 1 ď i ď s and k i`1 " k i `αi´s if s `1 ď i ď s `r ´1.
In particular, we have k r`s `αr " a. According to [GP00] Proposition 3.4.7, the conjugacy classes in W a are labelled by bipartitions of a, and a representative of minimal length of the conjugacy class corresponding to the bipartition pα, βq is given by w α,β . Thus, the irreducible representations of W a can be labelled by bipartitions of a as well. An explicit construction of these irreducible representations is given in [GP00] 5.5. We will not recall it, however we may again give a method to compute the character values, similar to the Murnaghan-Nakayama formula. The character of the irreducible representation of W a associated in loc. cit. to the bipartition pα, βq of a will be denoted χ α,β . If pγ, δq is another bipartition of a, we denote by χ α,β pγ, δq the value of the character χ α,β on the conjugacy class of W a labelled by pγ, δq.

One can think of a bipartition pα, βq of a as an ordered pair of two Young diagrams of combined size a. A border strip of a bipartition pα, βq is a border strip either of the partition α or of β. The height of a border strip is defined in the same way.

Theorem ([GP00] Theorem 10.3.1). Let pα, βq and pγ, δq be two bipartitions of a. If γ " H, let ǫ " 1 and let x be the last integer in the partition γ. If γ " H, let ǫ " ´1 and let x be the last integer of the partition δ. We have

χ α,β pγ, δq " ÿ S p´1q htpSq ǫ f S χ pα,βqzS ppγ, δqzxq,
where S runs over the set of all border strips of size x in the bipartition pα, βq, such that removing S from pα, βq gives again a pair of Young diagrams. Here, the pair of Young diagrams pα, βqzS is the one obtained after removing S, and pγ, δqzx is the bipartition obtained by removing x from pγ, δq. Eventually, the integer f S is 0 if S is a border strip of α, and it is 1 if S is a border strip of β.

Applying this formula in successions results in the value of χ pα,βq pγ, δq. In particular, one sees that χ paq,H is the trivial character and χ H,p1 a q is the signature character of W a . We illustrate the computations with pα, βq " pp3, 1 2 q, p4, 2qq and pγ, dq " pp4q, p5, 2qq. There is only elligible border strip of size 4 in the pair of diagrams pα, βq, as marked below.

, ˆˆ3

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

This border strip S has height 1. It was taken in the diagram of β so f S " 1. Since γ " H we have ǫ " 1. Applying the formula, we obtain χ p3,1 2 q,p4,2q pp4q, p5, 2qq " ´χp3,1 2 q,p1 2 q pH, p5, 2qq.

We are now looking for border strips of size 2 in the pair of diagrams of the bipartition p3, 1 2 q, p1 2 q. Three of them are eligible, as marked below.

, ˆand ˆˆ, and ˆ,

These three border strips have respective heights 1, 0 and 1. The corresponding values of f S are respectively 1, 0 and 0. Moreover, the partition γ is now empty so ǫ " ´1. The formula gives χ p3,1 2 q,p1 2 q pH, p5, 2qq " χ p3,1 2 q,H pH, p5qq `χp1 3 q,p1 2 q pH, p5qq ´χp3q,p1 2 q pH, p5qq.

In the bipartitions pp1 3 q, p1 2 qq and pp3q, p1 2 qq there is no border strip of size 5 at all. Thus, the formula tells us that the corresponding character values are 0. On the other hand, the bipartition pp3, 1 2 q, Hq consists of a single border strip of size 5 and height 2. The formula gives χ p3,1 2 q,H pH, p5qq " χ H " 1.

Putting things together, we deduce that χ p3,1 2 q,p4,2q pp4q, p5, 2qq " ´1.

2.3.4

We may now describe the horizontal arrows in 2.3.1 Theorem for the unitary group.

To do this, we need an alternate labelling of the irreducible unipotent representations of the unitary group. We refer to [START_REF] Fong | Brauer trees in classical groups[END_REF] for the details.

The new labelling of the irreducible unipotent representations of U n pqq involves triples of the form p∆ x , α, βq where x is a nonnegative integer such that n " 2a`x px`1q 2 for some integer a ě 0, and where pα, βq is a bipartition of a. The corresponding representation will be denoted ρ ∆x,α,β . With this labelling, the unipotent Harish-Chandra series EpU n pqq, pL x , ρ x qq consists precisely of all the representations ρ ∆x,α,β with pα, βq varying over all bipartitions of a. The bijection ZEpU n pqq, pL x , ρ x qq " Ý Ñ ZIrrpW Unpqq pL x , ρ x qq involved in the Comparison theorem simply sends ρ ∆x,α,β to χ α,β . Here, we made use of the identification W Unpqq pL x , ρ x q » W a as in ??. More generally, if M is a standard Levi complement in U n pqq containing L x , we may write M » U b pqq ˆGL a 1 pq 2 q ˆ. . . ˆGL ar pq 2 q where n " 2pa 1 `. . . `ar q `b and b ě xpx`1q

2

. The irreducible unipotent representations of M in the Harish-Chandra series EpM, pL x , ρ x qq are those of the form ρ ∆x,α,β b ρ GL λ 1 b . . . b ρ GL λr where λ i is a partition of a i for 1 ď i ď r and pα, βq is a bipartition of the integer c :" 1 2 ´b ´xpx`1q 2 ¯. On the other hand, the relative Weyl group W M pL x , ρ x q can be identified with the subgroup of W Unpqq pL x , ρ x q » W a isomorphic to the product W c ˆSa 1 ˆ. . . ˆSar (note that c `a1 `. . . `ar " a). With the notations of 2.3.2, the W c -component is generated by the elements θ, σ 1 , . . . σ c´1 , the S a 1 -component by the elements σ c`1 . . . , σ c`a 1 ´1, and so on. Irreducible characters of W M pL x , ρ x q have the shape

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q χ α,β b χ λ 1 b . . . b χ λr where pα, βq is a bipartition of c and λ i is a partition of a i for 1 ď i ď r.

Then, according to [FS90] (4.2), the bijection ZEpM, pL x , ρ x qq

" Ý Ñ ZIrrpW M pL x , ρ x qq involved in the Comparison theorem in 2.3.1 sends ρ ∆x,α,β b ρ GL λ 1 b . . . b ρ GL λr to χ α,β b χ λ 1 b . . . b χ λr .

2.3.5

We explain how the two different labellings of the irreducible unipotent representations of U n pqq are related. To do this, one needs the notion of 2-quotient. For the following definitions, we allow partitions to have 0 terms at the end. Thus, let us write λ " pλ 1 ě . . . ě λ r q with λ r ě 0. The β-set of λ is the sequence of decreasing nonnegative integers β i :" λ i `r ´i for 1 ď i ď r. Mapping a partition λ to its β-set gives a bijection between the set of partitions having r terms and the set of decreasing sequences of nonnegative integers of length r. The inverse mapping sends a sequence pβ 1 ą . . . ą β r ě 0q to the partition λ given by λ i " β i `i´r.

Let λ be a partition of n as above, and let β be its β-set. We let β even (resp. β odd ) be the subsequence consisting of all even (resp. odd) integers of β. Then, we define the following sequences.

β 0 :" ˆβi 2 ˇˇˇβ i P β even ˙β1 :" ˆβi ´1 2 ˇˇˇβ i P β odd Ṫhe
sequences β 0 and β 1 are the β-sets of two partitions, which we call µ 0 and µ 1 respectively. Then, the 2-quotient of λ is the bipartition pµ 0 , µ 1 q if r is odd, and pµ 1 , µ 0 q if r is even. We note that the ordering of µ 0 and µ 1 in the 2-quotient may vary in the literature. Here, we followed the conventions of [START_REF] Fong | Brauer trees in classical groups[END_REF] section 1. A different ordering is used in [Jam84] 2.7.29. In loc. cit. Theorem 2.7.37, another construction of the 2-quotient using Young diagrams is proposed. Let λ 1 be another partition which differs from λ only by 0 terms at the end. While the β-sets of λ and λ 1 are not the same, the resulting 2-quotients are equal up to 0 terms at the end of the partitions. Thus, from now on we identify all partitions differing only from 0 terms by removing all of them. The 2-quotient of a partition is then well-defined.

Theorem ([Jam84] Theorem 2.7.30). A partition λ is uniquely characterized by the data of its 2-core ∆ x and its 2-quotient pλ 0 , λ 1 q. Moreover, the lengths of these partitions are related by the equation

|λ| " |∆ x | `2p|λ 0 | `|λ 1 |q and |∆ x | " xpx`1q 2 .
For instance, the 2-quotient of the partition λ " p3 2 , 2 2 , 1q is p2 2 , 1q. Recall that the 2-core of λ is ∆ 1 . Thus, the equation on the lengths of the partitions is satisfied, as we have 11 " 1`2p4`1q.

We may now relate the two labellings tρ U λ u and tρ ∆x,α,β u of the irreducible unipotent representations of U n pqq together.

Proposition ([FS90] Appendix). Let λ be a partition of n. Denote by ∆ y its 2-core and by pλ 0 , λ 1 q its 2-quotient. On the other hand, let x ě 0 be such that n " 2a `xpx`1q 2 for some a ě 0 and let pα, βq be a bipartition of a. Then the irreducible representations ρ U λ and ρ ∆x,α,β are equivalent if and only if x " y and pλ 0 , λ 1 q " pα, βq if x is even or pλ 0 , λ 1 q " pβ, αq if x is odd.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q For instance, for λ " p3 2 , 2 2 , 1q the representation ρ U λ is equivalent to ρ ∆ 1 ,p1q,p2 2 q .

2.3.6

In order to apply the comparison theorem 2.3.1 for unitary groups, it remains to understand how to compute inductions in Coxeter groups of type B. Such computations are carried out in [GP00] Section 6.1. It turns out that we will only need one specific case of such inductions, and the corresponding method is known as the Pieri rule for groups of type B.

Proposition ([GP00] 6.1.9). Let a ě 1 and consider r, s ě 0 such that r `s " a. We think of the group W r ˆSs as a subgroup of W a as in 2.3.4.

-Let pα, βq be a bipartition of r. Then the induced character Ind Wa WrˆSs `χpα,βq b χ psq ȋs the multiplicity-free sum of all the characters χ γ,δ such that for some 0 ď k ď s, the Young diagram of γ (resp. δ) can be obtained from that of α (resp. β) by adding k boxes (resp. s ´k boxes) so that no two of them lie in the same column.

-Let pγ, δq be a bipartition of a. The restricted character

Res Wa

Wr pχ γ,δ q is the multiplicity-free sum of all the characters χ pα,βq such that for some 0 ď k ď s, the Young diagram of α (resp. β) can be obtained from that of γ (resp. δ) by deleting k boxes (resp. s ´k boxes) so that no two of them lie in the same column.

We will use this rule on concrete examples in the sections that follow.

2.4

The cohomology of the Coxeter variety for the unitary group 2.4.1 In this section, we describe the cohomology of the Coxeter varieties for the unitary groups in odd dimension in terms of the classification of unipotent representations that we recalled in the previous section. The cohomology groups are entirely understood by the work of Lusztig in [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF].

Let t ě 0. The Coxeter variety for U 2t`1 pqq is the Deligne-Lusztig variety X H pcoxq, where cox is any Coxeter element of the Weyl group W » S 2t`1 . Recall that a Coxeter element is a permutation which can be written as the product, in any order, of exactly one simple reflexion for each F -orbit on S. The variety X H pcoxq does not depend on the choice of the Coxeter element. It is defined over F q 2 and is equipped with commuting actions of both U 2t`1 pqq and F 2 .

Notation. We write X t " X H pcoxq for the Coxeter variety attached to the unitary group U 2t`1 pqq. We also write H ' c pX t q instead of H ' c pX H pcoxq b F, Q ℓ q, where ℓ " p.

We first recall known facts on the cohomology of X t from Lusztig's work.

Theorem ([Lus76]

). The following statements hold.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

(1) The variety X t has dimension t and is affine. The cohomology group H t`i c pX t q is zero unless 0 ď i ď t.

(2) The Frobenius F 2 acts in a semisimple manner on the cohomology of X t .

(3) The group H 2t c pX t q is 1-dimensional, the unitary group U 2t`1 pqq acts trivially whereas F 2 has a single eigenvalue q 2t . (4) The group H t`i c pX t q for 0 ď i ă t is the direct sum of two eigenspaces of F 2 , for the eigenvalues q 2i and ´q2i`1 . Each eigenspace is an irreducible unipotent representation of U 2t`1 pqq.

(5) If 0 ď a ď 2t, the dimension of the eigenspace of p´qq a inside the sum ř iě0 H t`i c pX t q is given by the formula

q p2t´aqp2t`1´aq 2 2t´a ź j"1
q a`j ´p´1q a`j q j ´p´1q j .

(6) The sum ř iě0 H t`i c pX t q is multiplicity-free as a representation of U 2t`1 pqq.

2.4.2

We wish to identify these unipotent representations of U 2t`1 pqq occuring in the cohomology of X t . To this purpose, we start by defining the following partitions. If 0 ď a ď 2t, we put λ t a :" p1 `a, 1 2t´a q. Note that λ t 0 " p1 2t`1 q and λ t 2t " p2t `1q.

Lemma. For 0 ď i ď t, the 2-core of λ t 2i is ∆ 1 and its 2-quotient is pp1 t´i q, piqq. For 0 ď i ă t, then the 2-core of λ t 2i`1 is ∆ 2 and its 2-quotient is ppiq, p1 t´i´1 qq.

In particular, according to 2.3.5 the irreducible unipotent representation ρ λ t 2i of U 2t`1 pqq is equivalent to the representation ρ ∆ 1 ,piq,p1 t´i q , and ρ λ t 2i`1 to ρ ∆ 2 ,piq,p1 t´i´1 q .

Proof. The Young diagram of the partition λ t a has the following shape.

. . .

. . .

The first row has an odd number of boxes when a is even, and an even number of boxes when a is odd. To compute the 2-core, one removes horizontal dominoes from the first row, right to left, and vertical dominoes from the first column, bottom to top. The process results in ∆ 1 when a is even and ∆ 2 when a is odd.

The partition λ t a has 2t `1 ´a non zero terms. Its β-set is given by the sequence β " p2t `1, 2t ´a, 2t ´a ´1, . . . , 1q.

Assume that a " 2i is even. Then the sequences β 0 and β 1 are given by β 0 " pt ´i, t ´i ´1, . . . 1q, β 1 " pt, t ´i ´1, t ´i ´2, . . . , 0q.

The sequence β 0 has length t ´i while β 1 has length t ´i `1. The associated permutations are then respectively µ 0 " p1 t´i q and µ 1 " piq. Since 2t `1 ´a is odd, the 2-quotient is given by pµ 0 , µ 1 q as claimed.

Assume now that a " 2i `1 is odd. Then the sequences β 0 and β 1 are given by β 0 " pt ´i ´1, t ´i ´2, . . . 1q, β 1 " pt, t ´i ´1, t ´i ´2, . . . , 0q.

The sequence β 0 has length t ´i ´1 while β 1 has length t ´i `1. The associated permutations are then respectively µ 0 " p1 t´i´1 q and µ 1 " piq. Since 2t `1 ´a is even, the 2-quotient is given by pµ 1 , µ 0 q as claimed.

2.4.3

We may now identify the irreducible unipotent representations occuring in the cohomology of the Coxeter variety X k .

Proposition. For 0 ď i ă t, the cohomology group of the Coxeter variety for the finite unitary group U 2t`1 pqq is given by

H t`i c pX t q " ρ λ t 2i ' ρ λ t 2i`1
with the first summand corresponding to the eigenvalue q 2i of F 2 and the second to ´q2i`1 . Moreover, H 2t c pX t q " ρ λ t 2t with eigenvalue q 2t . Before going to the proof, one may notice that the statement is consistent with the dimensions. Indeed, the formula given in 2.4.1 Theorem (5) coincides with the hook formula for the degree of the representation ρ U λ t a given in 2.2.5 Proposition.

Proof. First, the statement on the highest cohomology group H 2t c pX t q follows from 2.4.1 Theorem (3). It is the only cohomology group in the case t " 0. We will prove the formula by induction on t. Let us now assume t ě 1 and that the proposition is known for t ´1. If 0 ď i ď t ´1, we know that H t`i c pX t q is the sum of two irreducible unipotent representations. So let us write H t`i c pX t q " ρ µ i ' ρ ν i where µ i and ν i are two partitions of 2t `1, and so that ρ µ i corresponds to the eigenvalue q 2i of F 2 whereas ρ ν i corresponds to ´q2i`1 . We consider the standard Levi complement L » GL 1 pq 2 q ˆU2t´1 pqq Ă U 2t`1 pqq. Let V denote the unipotent radical of the standard parabolic subgroup containing L. According to [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] Corollary 2.10, one can build a geometric isomorphism between the quotient variety X t {V and the product of the Coxeter variety for L and of a copy of G m . Even though this geometric isomorphism is not L-equivariant, Lusztig proves that the induced map on cohomology is Lequivariant. By a discussion similar to that in 2.1.10, the Coxeter variety for L is isomorphic to the Coxeter variety X t´1 for U 2t´1 pqq. We write ˚Rt t´1 for the composition of the Harish-Chandra restriction from U 2t`1 pqq to L, with the usual restriction from L to the subgroup U 2t´1 pqq. For any nonnegative integer i, the U 2t´1 pqq, F 2 -equivariant induced map on the cohomology is an isomorphism

˚Rt t´1 `Ht`i c pX t q ˘» H t´1`i c pX t´1 q ' H t´1`pi´1q c pX t´1 qp1q. (˚˚)
Here, p1q denotes the Tate twist (the action of F 2 on a twist M pnq is obtained from the action on the space M by multiplication with q 2n ). The right-hand side of this identity is given by the induction hypothesis. Let us look at the left-hand side. We fix 0 ď i ď t´1 and we denote by p∆ x , α, βq and by p∆ y , γ, δq the alternative labelling of the representations ρ µ i and ρ ν i respectively as introduced in 2.3.4 and 2.3.5. By the Howlett-Lehrer comparison theorem for restriction in 2.3.1 and by the Pieri rule in 2.3.6, we know that the restriction ˚Rt t´1 pρ ∆x,α,β q is the multiplicity-free sum of all the representations ρ ∆x,α 1 ,β 1 where the bipartition pα 1 , β 1 q can be obtained from pα, βq by removing exactly one box, of either α or β. The similar description also holds for ˚Rt t´1 `ρ∆y,γ,δ ˘.

By using 2.4.2 Lemma and the induction hypothesis, we may write down the identity (˚˚) explicitely. Moreover, as it is F 2 -equivariant we can identify the components corresponding to the same eigenvalues on both sides. We distinguish 4 different cases depending on the values of t and i.

-Case t " 1. We only need to consider i " 0. On the right-hand side of (˚˚), the second term is 0 because t´1`pi´1q " ´1 ă 0. On the other hand, the first term is ρ λ 0 0 » ρ ∆ 1 ,H,H and it corresponds to the eigenvalue p´qq 0 " 1. By identifying the eigenspaces, we have ˚R1 0 pρ ∆x,α,β q » ρ ∆ 1 ,H,H and ˚R1 0 `ρ∆y,γ,δ ˘" 0. The second equation implies that there is no box to remove from γ nor from δ. Thus, γ " δ " H. The value of y is given by the relation 2t `1 " 3 " 2p0 `0q `ypy`1q 2 , that is y " 2. This corresponds to the partition ν 0 " λ 1 1 . We notice in passing that the representation ρ ν 0 is the unique unipotent cuspidal representation of U 3 pqq. As for µ 0 , the equation ˚R1 0 pρ ∆x,α,β q » ρ ∆ 1 ,H,H tells us that there is only one removable box from pα, βq. After removal of this box, both partitions are empty. Thus, we deduce that x " 1 and pα, βq " p1, Hq or pH, 1q. This corresponds respectively to µ 0 " λ 1 2 or µ 0 " λ 1 0 . That is, ρ µ 0 is either the trivial or the Steinberg representation of U 3 pqq. We can deduce which one it is by comparing the degree of the representations with the formula of 2.4.1 Theorem (5). According to this formula, the dimension of the eigenspace for p´qq 0 is q 3 . This is precisely the degree of the Steinberg representation ρ λ 1 0 as given by the hook formula in 2.2.5 Proposition, and it excludes the possibility of ρ µ 0 being trivial. Thus, we have µ 0 " λ 1 0 as claimed. From now, we assume t ě 2.

-Case i " 0. On the right-hand side of (˚˚), the second term is 0 because t ´1 `pi ´1q "

t ´2 ă t ´1. The first term is ρ λ t´1 0 ' ρ λ t´1 1 » ρ ∆ 1 ,H,p1 t´1 q ' ρ ∆ 2 ,H,p1 t´2 q .
Identifying the eigenspaces, we have ˚Rt t´1 pρ ∆x,α,β q » ρ ∆ 1 ,H,p1 t´1 q and ˚Rt t´1 `ρ∆y,γ,δ ˘» ρ ∆ 2 ,H,p1 t´2 q . We deduce that x " 1 and y " 2. Moreover, it also follows that there is only one removable box in pα, βq and in pγ, δq. After removal, we should obtain respectively pH, p1 t´1 qq and pH, p1 t´2 qq. The only possibility is that pα, βq " pH, p1 t qq and pγ, δq " pH, p1 t´1 qq. This corresponds to µ 0 " λ t 0 and ν 0 " λ t 1 as claimed. -Case i " t ´1. On the right-hand side of (˚˚), the first term is

ρ λ t´1 2pt´1q » ρ ∆ 1 ,pt´1q,H and the second term is ρ λ t´1 2pt´2q 'ρ λ t´1 2pt´2q`1 » ρ ∆ 1 ,pt´2q,p1q 'ρ ∆ 2 ,
pt´2q,H . Identifying the eigenspaces while taking the Tate twist into account, we have ˚Rt t´1 pρ ∆x,α,β q » ρ ∆ 1 ,pt´1q,H 'ρ ∆ 1 ,pt´2q,p1q and ˚Rt t´1 `ρ∆y,γ,δ ˘» ρ ∆ 2 ,pt´2q,H . We deduce that x " 1 and y " 2. Moreover, there are two removable boxes in pα, βq and only one removable box in pγ, δq. After removal of one of the two boxes in pα, βq, we can get either ppt ´1q, Hq or ppt ´2q, p1qq ; and after removal of the box in pγ, δq we obtain ppt ´2q, Hq. The only possibility is that pα, βq " ppt ´1q, p1qq and pγ, δq " ppt ´1q, Hq. This corresponds to µ t´1 " λ t 2pt´1q and ν t´1 " λ t 2pt´1q`1 as claimed. -Case 1 ď i ď t ´2. On the right-hand side of (˚˚), the first term is

ρ λ t´1 2i ' ρ λ t´1 2i`1 » ρ ∆ 1 ,piq,p1 t´1´i q ' ρ ∆ 2 ,piq,p1 t´2´i q . The second term is ρ λ t´1 2pi´1q ' ρ λ t´1 2pi´1q`1 » ρ ∆ 1 ,pi´1q,p1 t´i q ' ρ ∆ 2 ,
pi´1q,p1 t´1´i q . Identifying the eigenspaces while taking the Tate twist into account, we have ˚Rt t´1 pρ ∆x,α,β q » ρ ∆ 1 ,piq,p1 t´1´i q ' ρ ∆ 1 ,pi´1q,p1 t´i q and ˚Rt t´1 `ρ∆y,γ,δ ˘» ρ ∆ 2 ,piq,p1 t´2´i q ' ρ ∆ 2 ,pi´1q,p1 t´1´i q . We deduce that x " 1 and y " 2. Moreover, there are exactly two removable boxes from pα, βq and from pγ, δq. After removal of one of the two boxes in pα, βq, we can get either ppiq, p1 t´1´i qq or ppi ´1q, p1 t´i qq ; and after removal of one of the two boxes in pγ, δq, we can get either ppiq, p1 t´2´i qq or ppi ´1q, p1 t´1´i qq. The only possibility is that pα, βq " ppiq, p1 t´i qq and pγ, δq " ppiq, p1 t´1´i qq. This corresponds to µ i " λ t 2i and ν i " λ t 2i`1 as claimed.

2.5

The cohomology of the variety X I pidq 2.5.1 We go on with the computation of the cohomology of the variety X I pidq. We use the same notations as in section 1. We first compute the cohomology of each Ekedahl-Oort stratum X It pw t q, before using the spectral sequence associated to the stratification to conclude. Recall that X I pidq has dimension d, is defined over F q 2 and is equipped with an action of J » U 2d`1 pqq. As before, we will write H ' c pX I pidqq as a shortcut for H ' c pX I pidq b F, Q ℓ q.

Theorem. The following statements hold.

(1) The cohomology group H i c pX I pidqq is zero unless 0 ď i ď 2d. There is an isomorphism H i c pX I pidqq » H 2d´i c pX I pidqq _ pdq which is equivariant for the actions of F 2 and of U 2d`1 pqq.

(2) The Frobenius F 2 acts like multiplication by p´qq i on H i c pX I pidqq.

(3) For 0 ď i ď d we have

H 2i c pX I pidqq " minpi,d´iq à s"0 ρ p2d`1´2s,2sq .
For 0 ď i ď d ´1 we have

H 2i`1 c pX I pidqq " minpi,d´1´iq à s"0 ρ p2d´2s,2s`1q .
Thus, in the cohomology of X I pidq all the representations associated to a Young diagram with at most 2 rows occur, and there is no other. Such a diagram has the following general shape.

. . .

. . .

We may rephrase the result by using the alternative labelling of the irreducible unipotent representations as in 2.3.5. The partition p2d`1´2s, 2sq has 2-core ∆ 1 and 2-quotient pH, pdś , sqq ; whereas the partition p2d ´2s, 2s `1q has 2-core ∆ 2 and 2-quotient ppd ´1 ´s, sq, Hq. Thus, according to 2.3.5 Proposition, we have

ρ p2d`1´2s,2sq » ρ ∆ 1 ,pd´s,sq,H , ρ p2d´2s,2s`1q » ρ ∆ 2 ,pd´1´s,sq,H .
In particular, all irreducible representations in the cohomology groups of even index belong to the unipotent principal series EpU 2d`1 pqq, pL 1 , ρ 1 qq, whereas all the ones in the groups of odd index belong to the Harish-Chandra series EpU 2d`1 pqq, pL 2 , ρ 2 qq.

Proof. Point (1) of the statement follows from a general property of the cohomology groups, namely Poincaré duality. It is due to the fact that X I pidq is projective and smooth. It also implies the purity of the Frobenius F 2 on the cohomology : we know at this stage that all eigenvalues of F 2 on H i c pX I pidqq have complex modulus q i under any choice of an isomorphism Q ℓ » C. We prove the points (2) and (3) by explicit computations. As in 2.4.2, we denote by λ t a the partition p1 `a, 1 2t´a q of 2t `1. Let 0 ď t ď d. For 0 ď a ď 2t we will write

R t a :" R U 2d`1 pqq L K t `ρGL pd´tq b ρ U λ t a ˘.
Recall that 2.1.9 Proposition gives an isomorphism between the Ekedahl-Oort stratum X It pw t q and the variety U 2d`1 pqq{U Kt ˆLK t X

L K t
It pw t q. It implies that the cohomology of the Ekedahl-Oort stratum is the Harish-Chandra induction of the cohomology of the Deligne-Lusztig variety X L K t It pw t q. According to 2.1.10, this cohomology is related to that of the Coxeter variety for U 2t`1 pqq. Combining with the formula of 2.4.3 Proposition, for 0 ď i ď t ´1 it follows that

H t`i c pX It pw t qq " R t 2i ' R t 2i`1 , H 2t c pX It pw t qq " R t 2t .
The representation R t a in this formula is associated to the eigenvalue p´qq a of F 2 .

We first compute R t a explicitely. By the combination of the Howhlett-Lehrer comparison theorem in 2.3.1 and the Pieri rule for groups of type B as in 2.3.6, one can compute the Harish-Chandra induction R t a by adding d ´t boxes to the bipartition corresponding to the representation ρ U λ t a with no two added boxes in the same column. Recall from 2.4.2 Lemma that the Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q representation ρ λ t 2i of U 2t`1 pqq is equivalent to the representation ρ ∆ 1 ,piq,p1 t´i q , and that ρ λ t 2i`1 is equivalent to ρ ∆ 2 ,piq,p1 t´1´i q . In order to illustrate the argument, let us say that we want to add N boxes to a bipartition of the shape as in the figure below, so that no two added boxes lie in the same column.

. . . , . . .

We will add N 1 boxes to the first diagram and N 2 to the second, where N " N 1 `N2 . In the first diagram, the only places where we can add boxes are in the second row from left to right, and at the end of the first row. Because no two added boxes must be in the same column, the number of boxes we add on the second row must be at most the number of boxes already lying in the first row. Of course, it must also be at most N 1 .

In the second diagram, the only places where we can add boxes are at the bottom of the first column and at the end of the first row. Because no two added boxes must be in the same column, we can only put up to one box at the bottom of the first column and all the remaining ones will align at the end of the first row. At the end of the process, we will obtain a bipartition of the following general shape. We colored in yellow the boxes that were already there before we added new ones. The box with a question mark may or may not be placed there.

We now make the result more precise, and write down exactly what the irreducible components of R t a are depending on the parity of a.

-For 0 ď i ď t, the representation R t 2i is the multiplicity-free sum of all the representations ρ ∆ 1 ,α,β where the bipartition pα, βq satisfies, for some 0 ď x ď d ´t,

$ & % α " pi `x ´s, sq for some 0 ď s ď minpx, iq, β " pd ´t ´x, 1 t´i q or pd ´t ´x `1, 1 t´i´1 q.

-For 0 ď i ď t´1, the representation R t 2i`1 is the multiplicity-free sum of all representations ρ ∆ 2 ,α,β where the bipartition pα, βq satisfies, for some 0 ď x ď d ´t, $ & % α " pi `x ´s, sq for some 0 ď s ď minpx, iq, β " pd ´t `1 ´x, 1 t´1´i q or pd ´t `2 ´x, 1 t´2´i q.

In our notations, we used the convention that the partitions p0q and p1 0 q are the empty partition H. The integer x corresponds to the number of boxes we add to the first partition. We notice that if i takes the maximal value, there is only one possibility for β that is respectively pd´t´xq in the first case and pd ´t `1 ´xq in the second case. Recall from 2.1.7 that the variety X I pidq is the union of the Ekedahl-Oort strata X It pw t q for 0 ď t ď d and the closure of the stratum for t is the union of all strata X Is pw s q for s ď t. At the level of cohomology, it translates into the following F 2 , U 2d`1 pqq-equivariant spectral sequence

E t,i 1 : H t`i c pX It pw t qq ùñ H t`i c pX I pidqq.
The first page of the sequence is drawn in the Figure 1, it has a triangular shape.

R d 2d R d´1 2d´2 R d 2d´2 ' R d 2d´1 . . . . . . R 2 4 . . . R d´1 4 ' R d´1 5 R d 4 ' R d 5 R 1 2 R 2 2 ' R 2 3 . . . R d´1 2 ' R d´1 3 R d 2 ' R d 3 R 0 0 R 1 0 ' R 1 1 R 2 0 ' R 2 1 . . . R d´1 0 ' R d´1 1 R d 0 ' R d 1 Figure 1:
The first page of the spectral sequence.

The representation R t a corresponds to the eigenvalue p´qq a of F 2 as before. The only eigenvalues of F 2 on the i-th row of the spectral sequence are q 2i and ´q2i`1 . In particular, the eigenvalues on two distinct rows are different. Since the differentials in deeper pages of the sequence map terms from different rows, their F 2 -equivariance implies that they vanish. Therefore, the sequence degenerates on the second page. Moreover, by the machinery of spectral sequences, for 0 ď k ď 2d there exists a filtration by U 2d`1 pqq ˆxF 2 y-modules on H k c pX I pidqq whose graded components are the terms of the second page lying on the anti-diagonal t `i " k. Since the group algebra Q ℓ rU 2d`1 pqqs is semi-simple, the filtration splits, meaning that H k c pX I pidqq is actually the direct sum of the graded components. The purity of H k c pX I pidqq then implies that all the terms of the second page lying on the anti-diagonal t `i " k, which are associated to an eigenvalue whose modulus is not equal to q k , must be zero. Therefore, the second page has the shape described in Figure 2. The Frobenius F 2 acts via q 2i on the term E i,i 2 , and via ´q2i`1 on the term E i`1,i 2 . Point (2) of the Theorem readily follows.

E d,d 2 E d´1,d´1 2 E d,d´1 2 . . . . . . E 1,1 2 E 2,1 2 0 . . . 0 E 0,0 2 E 1,0 2 0 0 . . . 0 Figure 2:
The second page of the spectral sequence.

By the previous computations, we understand precisely all the terms in the first page of the spectral sequence. The key observation to compute the second page is that two terms on the first page which lie on the same row, but are separated by at least 2 arrows, do not have any irreducible component in common. We make the argument more precise in the following two paragraphs, distinguishing the cohomology groups of even and odd index.

We first compute the cohomology group H 2t c pX I pidqq for 0 ď t ď d. We look at the following portion of the first page

R t 2t R t`1 2t ' R t`1 2t`1 R t`2 2t ' R t`2 2t`1 .
By extracting the eigenspaces corresponding to q 2t , we actually have the following sequence

R t 2t R t`1 2t R t`2 2t u v
.

The representation R t 2t is the sum of all the representations ρ ∆ 1 ,α,β where for some 0 ď x ď d´t and for some 0 ď s ď minpx, tq, we have α " pt `x ´s, sq and β " pd ´t ´xq. The representation R t`1 2t is the sum of all the representations ρ ∆ 1 ,α 1 ,β 1 where for some 0 ď x 1 ď d ´t ´1 and for some 0 ď s ď minpx 1 , tq, we have α 1 " pt `x1 ´s, sq and β 1 " pd ´t ´x1 q or pd ´t ´x1 ´1, 1q. The quotient space Kerpvq{Impuq is isomorphic to the eigenspace of q 2t in E t`1,t 2 , which is zero. Besides, in the representation R t`2 2t all the irreducible components have the shape ρ ∆ 1 ,α 2 ,β 2 with β 2 a partition of length 2 or 3. In particular, all the representations ρ ∆ 1 ,α 1 ,β 1 of R t`1 2t with β 1 a partition of length 1 automatically lie inside Kerpvq " Impuq. Such representations correspond to all the irreducible components ρ ∆ 1 ,α,β of R t 2t having x " d ´t. Thus, none of them lies in Kerpuq » E t,t 2 . The remaining components of R t 2t are those having x " d ´t, and they do not occur in the codomain of u so that they lie in Kerpuq. By the previous argument, they must form the whole of Kerpuq. Thus, we have proved that

E t,t 2 » H 2t c pX I pidqq » Kerpuq " minpt,d´tq à s"0 ρ ∆ 1 ,pt´s,sq,H
and it coincides with the formula of point (3).

We now compute the cohomology group H 2t`1 c pX I pidqq for 0 ď t ď d ´1. We look at the following portion of the first page

R t 2t R t`1 2t ' R t`1 2t`1 R t`2 2t ' R t`2 2t`1 R t`3 2t ' R t`3 2t`1 .
By extracting the eigenspaces corresponding to ´q2t`1 , we actually have the following sequence

0 R t`1 2t`1 R t`2 2t`1 R t`3 2t`1 u v .
The representation R t`1 2t`1 is the sum of all the representations ρ ∆ 2 ,α,β where for some 0 ď x ď d ´t ´1 and for some 0 ď s ď minpx, tq, we have α " pt `x ´s, sq and β " pd ´t ´xq. The representation R t`2 2t`1 is the sum of all the representations ρ ∆ 2 ,α 1 ,β 1 where for some 0 ď x 1 ď d ´t ´2 and for some 0 ď s ď minpx 1 , tq, we have α 1 " pt `x1 ´s, sq and β 1 " pd ´t ´1 ´x1 , 1q or pd ´t ´x1 q. The quotient space Kerpvq{Impuq is isomorphic to the eigenspace of ´q2t`1 in E t`2,t 2 , which is zero. Besides, in the representation R t`3 2t`1 all the irreducible components have the shape ρ ∆ 2 ,α 2 ,β 2 with β 2 a partition of length 2 or 3. In particular, all the representations ρ ∆ 2 ,α 1 ,β 1 of R t`2 2t`1 with β 1 a partition of length 1 automatically lie inside Kerpvq » Impuq. Such representations correspond to all the irreducible components ρ ∆ 2 ,α,β of R t`1 2t`1 having x " d ´t ´1. Thus, none of them lies in Kerpuq » E t`1,t 2 . The remaining components of R t`1 2t`1 are those having x " d ´t ´1, and they do not occur in the codomain of u so that they lie in Kerpuq. By the argument above, they must form the whole of Kerpuq. Thus, we have proved that

E t`1,t 2 » H 2t`1 c pX I pidqq » Kerpuq " minpd´t´1,tq à s"0 ρ ∆ 2 ,pt´1´s,sq,H
and one may check that it coincides with the formula of point (3).

3 On the cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature p1, n ´1q

Notations

Throughout this section, we fix an integer n ě 1 and we write m :" t n´1 2 u so that n " 2m `1 or 2pm `1q according to whether n is odd or even. We also fix an odd prime number p. If k is a perfect field of characteristic p, we denote by W pkq the ring of Witt vectors and by W pkq Q its fraction field, which is an unramified extension of Q p . We denote by σ k : x Þ Ñ x p the Frobenius of Galpk{F p q, and we use the same notation for its (unique) lift to GalpW pkq Q {Q p q. If k 1 {k is a perfect field extension then pσ k 1 q |k " σ k , so we can remove the subscript and write σ unambiguously instead. If q " p e is a power of p, we write F q for the field with q elements. In the special case where q " p 2 , we also use the alternative notation Z p 2 " W pF p 2 q and Q p 2 " W pF p 2 q Q . We fix an algebraic closure F of F p .

The Bruhat-Tits stratification on the PEL unitary

Rapoport-Zink space of signature p1, n ´1q

3.1.1 The PEL unitary Rapoport-Zink space M of signature p1, n ´1q

3.1.1.1 In [VW11]
, the authors introduce the PEL unitary Rapoport-Zink space M of signature p1, n ´1q as a moduli space, classifying the deformations of a given p-divisible group equipped with additional structures. We briefly recall the construction. Let Nilp denote the category of schemes over Z p 2 where p is locally nilpotent. For S P Nilp, a unitary p-divisible group of signature p1, n ´1q over S is a triple pX, ι X , λ X q where -X is a p-divisible group over S.

ι X : Z p 2 Ñ EndpXq is a Z p 2 -action on X such that the induced action on its Lie algebra satisfies the signature p1, n ´1q condition: for every a P Z p 2 , the characteristic polynomial of ι X paq acting on LiepXq is given by pT ´aq 1 pT ´σpaqq n´1 P Z p 2 rT s Ă O S rT s.

λ X : X " Ý Ñ t X is a Z p 2 -linear polarization where t X denotes the Serre dual of X.

The Z p 2 -linearity of λ X is with respect to the Z p 2 -actions ι X and the induced action ιt X on the dual. A specific example of unitary p-divisible group over F p 2 is given in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 2.4 by means of covariant Dieudonné theory. We denote it by pX, ι X , λ X q and call it the standard unitary p-divisible group. The p-divisible group X is superspecial. The following set-valued functor M defines a moduli problem classifying deformations of X by quasi-isogenies. More precisely, for S P Nilp the set MpSq consists of all isomorphism classes of tuples pX, ι X , λ X , ρ X q such that Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

-pX, λ X , ρ X q is a unitary p-divisible group of signature p1, n ´1q over S.

ρ X : X ˆS S Ñ X ˆFp 2 S is a Z p 2 -linear quasi-isogeny compatible with the polarizations, in the sense that t ρ X ˝λX ˝ρX is a Q p -multiple of λ X .

In the second condition, S denotes the special fiber of S. By [RZ96] Corollary 3.40, this moduli problem is represented by a separated formal scheme M over SpfpZ p 2 q, called a Rapoport-Zink space. It is formally locally of finite type, and because the associated PEL datum is unramified it is also formally smooth over Z p 2 . The reduced special fiber of M is the reduced F p 2 -scheme M red defined by the maximal ideal of definition. By loc. cit. Proposition 2.32, each irreducible component of M red is projective. The geometry of the special fiber has been thoroughly described in [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] and [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF], and we recall some of their constructions.

3.1.1.2 Rational points of M over a perfect field extension k of F p 2 can be understood in terms of semi-linear algebra by means of Dieudonné theory. We denote by M pXq the Dieudonné module of X, this is a free Z p 2 -module of rank 2n. We denote by N pXq :" M pXq b Q p 2 its isocrystal. By construction, the Frobenius and the Verschiebung agree on N pXq. In particular, we have F 2 " p ¨id on the isocrystal. The Z p 2 -action ι X induces a Z{2Z-grading M pXq " M pXq 0 ' M pXq 1 as a sum of two free Z p 2 -modules of rank n. The same goes for the isocrystal N pXq " N pXq 0 ' N pXq 1 where N pXq i " M pXq i b Q p 2 for i " 0, 1. The polarization λ X induces a perfect σ-symplectic form on N pXq which stabilizes the lattice M pXq and for which F is selfadjoint. Compatibility with ι X implies that the pieces N pXq i are totally isotropic for i " 0, 1 and dual of each other. Moreover, the Frobenius F is then 1-homogeneous with respect to this grading. As in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 2.6, it is possible to modify the symplectic pairing so that it restricts to a non-degenerate Q p 2 -valued σ-hermitian form t¨, ¨u on N pXq 0 .

Notation. From now on, we will write V :" N pXq 0 and M :" M pXq 0 .

Then V is a Q p 2 -hermitian space of dimension n, and M is a given Z p 2 -lattice, ie. a Z p 2submodule containing a basis of V. Given two lattices M 1 and M 2 , the notation M 1 d Ă M 2 means that M 1 Ă M 2 and the quotient module M 2 {M 1 has length d. The integer d is called the index of M 1 in M 2 , and is denoted d " rM 2 : M 1 s. We have 0 ď d ď n. Given a lattice M Ă V, the dual lattice is denoted M _ . It consists of all the vectors v P V such that tv, M u Ă Z p 2 . Then, by construction the lattice M satisfies

pM _ 1 Ă M n´1 Ă M _ .
The existence of such a lattice M in V implies that the σ-hermitian structure on V is isomorphic to any one described by the following two matrices

T odd :" A 2m`1 , T even :" ¨Am 1 0 0 p A m ‹ ‹ ‹ ' .
Here, A k denotes the k ˆk matrix with 1's in the antidiagonal and 0 everywhere else.

Proposition ([Vol10] 1.15). There exists a basis of V such that t¨, ¨u is represented by the matrix T odd is n is odd and by T even if n is even.

3.1.1.3 A Witt decomposition on V is a set tL i u iPI of isotropic lines in V such that the following conditions are satisfied:

-For every i P I, there is a unique i 1 P I such that tL i , L i 1 u " 0.

-The sum of the L i 's is direct.

-The orthogonal in V of the direct sum of the L i 's is an anisotropic subspace of V.

Because each line L i is isotropic, in the first condition one necessarily has pi 1 q 1 " i and i " i 1 . As a consequence, the cardinality of the index set I is an even number #I " 2wpVq. The integer w " wpVq is called the Witt index of V and it does not depend on the choice of a Witt decomposition. We write L an for the orthogonal of the direct sum of the L i 's. The dimension of L an is n an :" n ´2w, therefore it is also independent on the choice of the Witt decomposition.

Given any Witt decomposition, one may always find vectors e i P L i such that te i , e j u " δ j,i 1 . Together with a choice of an orthogonal basis for L an , these vectors define a basis of V which is said to be adapted to the Witt decomposition. For any i P I, the direct sum L i ' L i 1 is isometric to the hyperbolic plane H. Therefore, we obtain a decomposition

V " wH ' L an .

We may always rearrange the index set so that I " t´w, . . . , ´1, 1, . . . , wu and for every i P I, we have tL i , L ´iu " 0. Thus, the i 1 associated to i by the first condition is ´i. Of course, this process is not unique as it relies on a choice of an ordering for the lines tL i u iPI . In this context, we write L 0 instead of L an .

3.1.1.4

We fix once and for all a basis e of V in which the hermitian form is represented by the matrix T odd or T even . In the case n " 2m `1 is odd, we will denote it e " pe ´m, . . . , e ´1, e an 0 , e 1 , . . . , e m q, and in the case n " 2pm `1q is even we will denote it e " pe ´m, . . . , e ´1, e an 0 , e an 1 , e 1 , . . . , e m q.

In this way, for every 1 ď s ď m the subspace generated by e ´s and e s is isomorphic to the hyperbolic plane H. Moreover, the vectors with a superscript ¨an generate an anisotropic subspace V an of V. The choice of such a basis gives a Witt decomposition

V " mH ' V an
consisting of an orthogonal sum of m copies of H and of the anisotropic subspace V an . In particular, the Witt index of V is m and we have n an " 1 or 2 depending on whether n is odd or even respectively.

3.1.1.5 Given a perfect field extension k of F p 2 , we denote by V k the base change V b Q p 2 W pkq Q . The form may be extended to V k by the formula tv b x, w b yu :" xy σ tv, wu P W pkq Q for all v, w P V and x, y P W pkq Q . The notions of index and duality for W pkq-lattices can be extended as well. We have the following description of the rational points of the Rapoport-Zink space.

Proposition ([Vol10] 1.10). Let k be a perfect field extension of F p 2 . There is a natural bijection between Mpkq " M red pkq and the set of lattices M in V k such that for some integer i P Z, we have

p i`1 M _ 1 Ă M n´1 Ă p i M _ .

3.1.1.6

There is a decomposition M " Ů iPZ M i into formal connected subschemes which are open and closed. The rational points of M i are those lattices M satisfying the relation above with the given integer i. Similarly, we have a decomposition into open and closed connected subschemes M red " Ů iPZ M i,red . In particular, the lattice M defined in the previous paragraph is an element of M 0 pF p 2 q. Not all integers i can occur though, as a parity condition must be satisfied by the following lemma.

Lemma ([Vol10] 1.7). The formal scheme M i is empty if ni is odd.

3.1.1.7 Let J " GUpVq be the group of unitary similitudes attached to V. It consists of all linear transformations g which preserve the hermitian form up to a unit cpgq P Q p , called the multiplier. One may think of J as the group of Q p -rational point of a reductive algebraic group. The space M is endowed with a natural action of J. At the level of points, the element g acts by sending a lattice M to gpM q. By [Vol10] 1.16, the action of g P J induces, for every integer i, an isomorphism M i " Ý Ñ M i`αpgq where αpgq is the p-adic valuation of the multiplier cpgq. This defines a continous homomorphism α : J Ñ Z where Z is given the discrete topology. According to 1.17 in loc. cit. the image of α is Z if n is even, and it is 2Z if n is odd. The center ZpJq of J consists of all the multiple of the identity. Therefore it can be identified with Q p2 . If λ P Q p2 , then cpλ¨idq " λσpλq " Normpλq P Q p , where Norm is the norm map relative to the quadratic extension Q p 2 {Q p . In particular, αpZpJqq " 2Z. Thus, the restriction of α to the center of J is surjective onto the image of α only when n is odd. When n is even, we define the following element

g 0 :" ¨Im 0 p 1 0 pI m ‹ ‹ ‹ '
where I m denotes the m ˆm identity matrix. Then g 0 P J and cpg 0 q " p so that αpg 0 q " 1. Moreover g 2 0 " p ¨id belongs to ZpJq. Let i and i 1 be two integers such that ni and ni 1 are even. Following [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] Proposition 1.18, we define a morphism ψ i,i 1 : M i Ñ M i 1 by sending, for any perfect field extension k{F p 2 , a point M P M i to

ψ i,i 1 pM q " $ & % p i 1 ´i 2 ¨M if i " i 1 mod 2. p i 1 ´i´1 2 g 0 ¨M if i ı i 1 mod 2.
This is well defined as the second case may only happen when n is even. We obtain the following proposition.

Proposition ([Vol10] 1.18). The map ψ i,i 1 is an isomorphism between M i and M i 1 . Moreover they are compatible with each other in the sense that if i, i 1 and i 2 are three integers such that ni, ni 1 and ni 2 are even, then we have

ψ i 1 ,i 2 ˝ψi,i 1 " ψ i,i 2 .
The same statement also holds for the special fiber M red . In particular, we have M i " H if and only if ni is even.

3.1.2

The Bruhat-Tits stratification of the special fiber M red 3.1.2.1 We now recall the construction of the Bruhat-Tits stratification on M red as in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF]. Let i be an integer such that ni is even. We define

L i :" tΛ Ă V a lattice | p i`1 Λ _ Ĺ Λ Ă p i Λ _ u.
If Λ P L i , we define its orbit type tpΛq :" rΛ : p i`1 Λ _ s. We also call it the type of Λ. In particular, the lattices in L i of type 1 are precisely the F p 2 -rational points of M i,red . By sending Λ to gpΛq, an element g P J defines a map L i Ñ L i`αpgq .

Proposition ([Vol10] Remark 2.3 and [VW11] Remark 4.1). Let i be an integer such that ni is even and let Λ P L i .

-The map L i Ñ L i`αpgq induced by an element g P J is an inclusion preserving, type preserving bijection. -We have 1 ď tpΛq ď n. Furthermore tpΛq is odd.

-The sets L i 's for various i's are pairwise disjoint.

Moreover, two lattices Λ, Λ 1 P Ů niP2Z L i are in the same orbit under the action of J if and only if tpΛq " tpΛ 1 q.

Proof. The first three points are proved in [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF]. Thus, we only explain the last statement. If Λ and Λ 1 are in the same J-orbit, because the action of J preserves the type we have tpΛq " tpΛ 1 q. For the converse, assume that Λ and Λ 1 have the same type. Let i and i 1 be the integers such that Λ P L i and Λ 1 P L i 1 . According to 3.1.1.7, we can always find g P J such that αpgq " i ´i1 . Hence, replacing Λ 1 by g ¨Λ1 we may assume that i " i 1 . Then the statement follows from [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] Remark 4.1.

We write L :" Ů niP2Z L i . For any integer i such that ni is even and any odd number t between 1 and n, there exists a lattice Λ P L i of orbit type t. Indeed, by fixing a bijection L i " Ý Ñ L 0 it is enough to find such a lattice for i " 0. Then, examples of lattices in L 0 of any type are given in 3.1.2.6 below.

3.1.2.2 Write t max :" 2m`1, so that the orbit type t of any lattice in L satisfies 1 ď t ď t max .

The following lemma will be useful later.

Lemma. Let i P Z such that ni is even, and let Λ P L. We have Λ _ P L if and only if either n is even, either n is odd and tpΛq " t max . If this condition is satisfied and n is even, then Λ _ P L ´i´1 and tpΛ _ q " n ´tpΛq. If on the contrary n is odd, then Λ _ P L ´i and tpΛ _ q " tpΛq.

Proof. First we prove the converse. We have the following chain of inclusions

p ´iΛ n´tpΛq Ă Λ _ tpΛq Ă p ´i´1 Λ.
If n is even, then ´npi `1q is also even and n ´tpΛq " 0. Since pΛ _ q _ " Λ, we deduce that Λ _ P L ´i´1 with orbit type n ´tpΛq. Assume now that n is odd and that tpΛq " t max " n. Then Λ _ " p ´iΛ P L ´i.

Let us now assume that Λ _ P L and that n is odd. Let i 1 P 2Z such that Λ _ P L i 1 . We have

Λ _ n´tpΛ _ q Ă p i 1 Λ n´tpΛq Ă p i 1 `iΛ _ , Λ _ tpΛq Ă p ´i´1 Λ tpΛ _ q Ă p ´i´i 1 ´2Λ _ ,
therefore ´2 ď i`i 1 ď 0. Since i`i 1 is even it is either ´2 or 0. If it were ´2, then we would have tpΛq " tpΛ _ q " 0 which is absurd. Therefore i`i 1 " 0, and we have n´tpΛq " n´tpΛ _ q " 0.

3.1.2.3 With the help of L i , one may construct an abstract simplicial complex B i . For s ě 0, an s-simplex of B i is a subset S Ă L i of cardinality s `1 such that for some ordering Λ 0 , . . . , Λ s of its elements, we have a chain of inclusions p i`1 Λ _ s Ĺ Λ 0 Ĺ Λ 1 Ĺ . . . Ĺ Λ s . We must have 0 ď s ď m for such a simplex to exist. We introduce J " SUpVq, the derived group of J. We consider the abstract simplicial complex BTp J, Q p q of the Bruhat-Tits building of J over Q p . A concrete description of this complex is given in [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF], while proving the following theorem.

Theorem ([Vol10] 3.5). The abstract simplicial complex BTp J, Q p q of the Bruhat-Tits building of J is naturally identified with B i for any fixed integer i such that ni is even. There is in particular an identification of L i with the set of vertices of BTp J, Q p q. The identification is J-equivariant.

Apartments in the Bruhat-Tits building BTp J, Q p q are in 1 to 1 correspondence with Witt decompositions of V. Let L " tL j u iPI be a Witt decomposition of V and let f " pf i q iPI \ B an be a basis of V adapted to the decomposition, where B an is an orthogonal basis of L an . Under the identification of BTp J, Q p q with B i , the vertices inside the apartment associated to L correspond to the lattices Λ P L i which are equal to the direct sum of Λ X L an and of the modules p r i Z p 2 f i for some integers pr i q iPI . The subset of L i consisting of all such lattices will be denoted A L i or, with an abuse of notations, A f i . We call such a set A L i the apartment associated to L in L i .

Remark. The set of vertices of the Bruhat-Tits building of J " GUpVq may then be identified with the disjoint union L of the L i 's. The subsets of lattices in a common apartment correspond to the sets A L :" Ů niP2Z A L i where L is some Witt decomposition of V. The set A L will be called the apartment associated to L.

We recall a general result regarding Bruhat-Tits buildings.

Proposition. Let i be an integer such that ni is even. Any two lattices Λ and Λ 1 in L i (resp. L) lie inside a common apartment A L i (resp. A L ) for some Witt decomposition L. Moreover, the action of the group J sends apartments to apartments. It acts transitively on the set tA L i u L . The same is true for J acting on the set tA L u L .

3.1.2.4

Recall the basis e of V that we fixed in 1.4. We will denote by Λpr ´m, . . . , r ´1, s, r 1 , . . . , r m q the Z p 2 -lattice generated by the vectors p r j e j for all j " ˘1, . . . , ˘m, by p s 0 e an 0 and if n is even, by p s 1 e an 1 too. Here, the r j 's are integers and s denotes either the integer s 0 if n is odd or the pair of integers ps 0 , s 1 q if n is even.

Proposition. Let i be an integer such that ni is even. Let pr j , sq be a family of integers as above. The corresponding lattice Λ " Λpr ´m, . . . , r ´1, s, r 1 , . . . , r m q belongs to L i if and only if the following conditions are satisfied -for all 1 ď j ď m, we have r ´j `rj P ti, i `1u, s 0 " t i`1 2 u, -if n is even, then s 1 " t i 2 u. Moreover, when that is the case the type of Λ is given by

tpΛq " 1 `2#t1 ď j ď m | r ´j `rj " iu.
Proof. The lattice Λ belongs to L i if and only if the following chain of inclusions holds:

p i`1 Λ _ Ĺ Λ Ă p i Λ _ .
The dual lattice Λ _ is equal to the lattice Λp´r m , . . . , ´r1 , s 1 , ´r´1 , . . . , ´r´m q, where s 1 " ´s0 when n is odd, and s 1 " p´s 0 , ´s1 ´1q when n is even. Thus, the inclusions above are equivalent to the following inequalities:

i ´r´j ď r j ď i `1 ´r´j , i ´s0 ď s 0 ď i `1 ´s0 , i ´1 ´s1 ď s 1 ď i ´s1 (if n is even).
This proves the desired condition on the integers r j 's and on s.

Let us now assume that Λ P L i . Its orbit type is equal to the index rΛ, p i`1 Λ _ s. This corresponds to the number of times equality occurs with the left-hand side in all the inequalities above. Of course, if the equality i ´r´j " r j occurs for some j, then it occurs also for ´j. Moreover, if i is even then the equality i ´s0 " s 0 occurs whereas i ´1 ´s1 " s 1 . On the contrary if i is odd, then the equality i ´1 ´s1 " s 1 occurs whereas i ´s0 " s 0 . Thus in all cases, only one of s 0 and s 1 contributes to the index. Putting things together, we deduce the desired formula.

3.1.2.5 We deduce the following corollary.

Corollary. The apartment A e i (resp. A e ) consists of all the lattices of the form Λ " Λpr ´m, . . . , r ´1, s, r 1 , . . . , r m q which belong to L i (resp. to L).

Proof. According to the previous proposition, it is clear that all lattices which belong to L i and are of the form Λpr ´m, . . . , r ´1, s, r 1 , . . . , r m q are elements of A e i . We shall prove the converse. Let Λ P A e i . By definition, there exists integers pr j q such that Λ " Λ X V an ' à 1ďjďm pp r ´j Z p 2 e ´j ' p r j Z p 2 e j q .

Write Λ 1 " Λ X V an . This is a lattice in V an which satisfies the chain of inclusions

p i`1 Λ 1 _ Ă Λ 1 Ă p i Λ 1 _ ,
where the duals are taken with respect to the restriction of t¨, ¨u to V an . Since V an is anisotropic, there is only a single lattice satisfying the chain of inclusions above. If we write a :" t i`1 2 u and b :" t i 2 u, it is given by p a Z p 2 e an 0 if n is odd, and by p a Z p 2 e an 0 ' p b Z p 2 e an 1 if n is even. Thus, it must be equal to Λ 1 and it concludes the proof.

3.1.2.6

We fix a maximal simplex in L 0 lying inside the apartment A e 0 . For 0 ď θ ď m we define Λ θ :" Λp0, . . . , 0 l jh n m , 0, 0, . . . , 0 l jh n θ , 1, . . . , 1 l jh n m´θ q.

Here, the 0 in the middle stands for p0, 0q in case n is even. The lattice Λ θ belongs to L 0 , its orbit type is 2θ `1 and together they fit inside the following chain of inclusions

pΛ _ 0 Ĺ Λ 0 Ă . . . Ă Λ m .
Thus, they form an m-simplex in L 0 .

3.1.2.7 Given a lattice Λ P L i , the authors of [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] define a subfunctor M Λ of M i,red classifying those p-divisible groups for which a certain quasi-isogeny, depending on Λ, is in fact an actual isogeny. In Lemma 4.2, they prove that it is representable by a projective scheme over F p 2 , and that the natural morphism M Λ ãÑ M i,red is a closed immersion. The schemes M Λ are called the closed Bruhat-Tits strata of M. Their rational points are described as follows.

Proposition ([VW11] Lemma 4.3). Let k be a perfect field extension of F p 2 , and let M P M i,red pkq. Then we have the equivalence

M P M Λ pkq ðñ M Ă Λ k :" Λ b Z p 2 W pkq.
The set of lattices satisfying the condition above was conjectured in [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] to be the set of points of a subscheme of M i,red , and it was proved in the special cases n " 2, 3. In [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF], the general argument is given by the construction of M Λ . The action of an element g P J on M red induces an isomorphism M Λ " Ý Ñ M g¨Λ .

3.1.2.8 Let Λ P L, we denote by J Λ the fixator of Λ under the action of J. If Λ " Λ θ for some 0 ď θ ď m, we will write J θ instead. These are maximal parahoric subgroups of J.

In unramified unitary similitude groups, maximal parahoric subgroups and maximal compact subgroups are the same. A general parahoric subgroup is an intersection J Λ 1 X . . . X J Λs where tΛ 1 , . . . , Λ s u is an s-simplex in L i for some i. Any parahoric subgroup is compact and open in J.

Let i be the integer such that Λ P L i . We define V 0 Λ :" Λ{p i`1 Λ _ and V 1 Λ :" p i Λ _ {Λ. Since pΛ Ă p ¨pi Λ _ and p ¨pi Λ _ Ă Λ, these are both F p 2 -vector space of dimensions respectively tpΛq and n ´tpΛq. Both spaces come together with a non-degenerate σ-hermitian form p¨, ¨q0 and p¨, ¨q1 with values in F p 2 , respectively induced by p ´it¨, ¨u and by p ´i`1 t¨, ¨u. If k is a perfect field extension of F p 2 and if ǫ P t0, 1u, we may extend the pairings to pV ǫ Λ q k " V ǫ Λ b F p 2 k by setting pv b x, w b yq ǫ :" xy σ pv, wq ǫ P k for all v, w P V ǫ Λ and x, y P k. If U is a subspace of pV ǫ Λ q k we denote by U K its orthogonal, that is the subspace of all vectors x P pV ǫ Λ q k such that px, U q ǫ " 0. Denote by J Λ the pro-unipotent radical of J Λ and write J Λ :" J Λ {J Λ . This is a finite group of Lie type, called the maximal reductive quotient of J Λ . We have an identification J Λ » GpUpV 0 Λ q ˆUpV 1 Λ qq, that is the group of pairs pg 0 , g 1 q where for ǫ P t0, 1u we have g ǫ P GUpV ǫ Λ q and cpg 0 q " cpg 1 q. Here, cpg ǫ q P F p denotes the multiplier of g ǫ . For 0 ď θ ď m and ǫ P t0, 1u, we will write V ǫ θ and J θ instead of V ǫ Λ θ and J Λ θ . A basis of V 0 θ is given by the images of the 2θ `1 vectors e ´θ . . . , e ´1, e an 0 , e 1 , . . . , e θ . As for V 1 θ , a basis is given by the images of the n ´2θ ´1 vectors p ´1e ´m, . . . , p ´1e ´θ´1 , e θ`1 , . . . , e m when n is odd, and in case n is even one must add the image of p ´1e an 1 to the basis.

3.1.2.9 Let Λ P L i where ni is even. We write tpΛq " 2θ `1. Let k be a perfect field extension of F p 2 . Let T be any W pkq-lattice in V k such that

p i`1 T _ 2θ 1 `1 Ă T Ă Λ k
where 0 ď θ 1 ď θ. Then T must contain p i`1 Λ _ k and rΛ k : T s " θ ´θ1 . We may consider T :"

T {p i`1 Λ _ k the image of T in V p0q Λ .
Then T is an F p 2 -subspace of dimension θ `θ1 `1. Moreover, one may check that p i`1 T _ " T K , therefore the subspace T contains its orthogonal.

These observations lead to the following proposition.

Proposition ([Vol10] 2.7). The mapping T Þ Ñ T defines a bijection between the set of W pkq-

lattices T in V k such that p i`1 T _ 2θ 1 `1 Ă T Ă Λ k and the set tU Ă pV 0 Λ q k | dim U " θ `θ1 `1 and U K Ă U u.
In particular taking θ 1 " 0, this set is in bijection with M Λ pkq.

Remark. Similarly, the set of W pkq-lattices

T such that Λ k Ă T n´2θ 1 ´1 Ă p i T _ for some θ ď θ 1 ď m is in bijection with tU Ă pV 1 Λ q k | dim U " n ´θ1 ´θ ´1 and U K Ă U u.
The bijection is given by T Þ Ñ T K where T :" T {Λ k Ă V p1q k . These sets can be seen as the k-rational points of some flag variety for GUpV p0q Λ q and GUpV p1q Λ q, which are special instances of Deligne-Lusztig varieties. This is accounted for in the next paragraph.

3.1.2.10 Let Λ P L. The action of J on the Rapoport-Zink space M restricts to an action of the parahoric subgroup J Λ on the closed Bruhat-Tits stratum M Λ . This action factors through the maximal reductive quotient J Λ » GpUpV 0 Λ q ˆUpV 1 Λ qq. This action is trivial on the normal subgroup tidu ˆUpV 1 Λ q Ă J Λ , thus it factors again through the quotient which is isomorphic to GUpV 0 Λ q.

Theorem ([VW11] Theorem 4.8). There is an isomorphism between M Λ and a certain "generalized" parabolic Deligne-Lusztig variety for the finite group of Lie type GUpV 0 Λ q, compatible with the actions. In particular, if tpΛq " 2θ `1 then the scheme M Λ is projective, smooth, geometrically irreducible of dimension θ.

We refer to [START_REF] Muller | Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature (1,n-1)[END_REF] Section 1 for the definition of Deligne-Lusztig varieties. In particular, the adjective "generalized" is understood according to loc. cit. The Deligne-Lusztig variety isomorphic to M Λ is introduced in [VW11] 4.5, and it is denoted by Y Λ there.

3.1.2.11

We now explain how the different closed Bruhat-Tits strata behave together.

Theorem ([VW11] Theorem 5.1). Let i P Z such that ni is even. Consider Λ and Λ 1 two lattices in L i . The following statements hold.

(1) The inclusion Λ Ă Λ 1 is equivalent to the scheme-theoretic inclusion M Λ Ă M Λ 1 . It also implies tpΛq ď tpΛ 1 q and there is equality if and only if Λ " Λ 1 . (2) The three following assertions are equivalent.

piq Λ X Λ 1 P L i . piiq Λ X Λ 1 contains a lattice of L i . piiiq M Λ X M Λ 1 " H.
If these conditions are satisfied, then M Λ X M Λ 1 " M ΛXΛ 1 , where we understand the left hand side as the scheme theoretic intersection inside M i,red .

(3) The three following assertions are equivalent

piq Λ `Λ1 P L i . piiq Λ `Λ1 is contained in a lattice of L i . piiiq M Λ , M Λ 1 Ă M r Λ for some r Λ in L i .
If these conditions are satisfied, then M Λ`Λ 1 is the smallest subscheme of the form M r

Λ containing both M Λ and M Λ 1 . (4) If k is a perfect field field extension of F p 2 then M i pkq " Ť ΛPL i M Λ pkq.
In essence, the previous statements explain how the stratification given by the M Λ mimics the combinatorics of the Bruhat-Tits building of J, hence the name.

On the maximal parahoric subgroups of J

3.1.3.1 In this section we give a few results that will be useful later regarding the maximal parahoric subgroups J Λ . First, we study their conjugacy classes. It starts with the following lemma.

Lemma. Let Λ, Λ 1 P L.

(i) The parahoric subgroup J Λ acts transitively on the set of apartments containing Λ.

(ii) We have J Λ " J Λ 1 if and only if there exists k P Z such that Λ " p k Λ 1 or Λ " p k Λ 1 _ .

Proof. The first point is a general fact from the theory of Bruhat-Tits buildings.

For the second point, the converse is clear. Indeed, if x P Q p2 then J xΛ " J Λ , and an element g P J fixes a lattice Λ if and only if it fixes its dual Λ _ . Now, let Λ, Λ 1 P L such that J Λ " J Λ 1 . Up to replacing Λ 1 by an appropriate lattice g ¨Λ1 , it is enough to treat the case Λ 1 " Λ θ for some 0 ď θ ď m. By 3.1.2.3 Proposition, we can find an apartment A L containing both Λ θ and Λ. By the first point, we can find g P J θ " J Λ which sends A L to A e . Therefore g ¨Λ " Λ belongs to A e . According to 3.1.2.5, we may write Λ " Λpr ´m, . . . , r ´1, s, r 1 , . . . , r m q for some integers pr j , sq. Let i be the integer such that Λ P L i . Then according to 3.1.2.4 we have -@1 ď j ď m, r ´j `rj P ti, i `1u.

s 0 " t i`1 2 u.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q -if n is even then s 1 " t i 2 u. For 1 ď j ď θ, let g j be the automorphism of V which exchanges e ´j and e j while fixing all the other vectors in the basis e. Then, from the definition of Λ θ we have g j P J θ . Therefore g j must fix Λ too, which implies that r ´j " r j . And for θ `1 ď j ď m, let g j be the automorphism sending e j to p ´1e ´j and e ´j to pe j while fixing all the other vectors in the basis e. Then again we have g j P J θ " J Λ which implies that r ´j " r j ´1. Assume first that i " 2i 1 is even. Combining the previous observations, we have r j " i 1 for all 1 ď j ď θ and r j " i 1 `1 for all θ `1 ď j ď m. Moreover we have s 0 " i 1 and if n is even, we have s 1 " i 1 . In other words, we have Λ " p i 1 Λ θ . Assume now that i " 2i 1 `1 is odd. This implies that n is even. Combining the previous observations, we have r j " i 1 `1 for all 1 ď j ď m. Moreover we have s 0 " i 1 `1 and if n is even, we have s 1 " i 1 . In other words, we have Λ " p i 1 `1Λ _ θ .

3.1.3.2

We may now describe the conjugacy classes of these maximal parahoric subgroups.

Corollary. Let Λ, Λ 1 P L.

(i) If n is odd, then tpΛq " tpΛ 1 q if and only if the associated maximal parahoric subgroups J Λ and J Λ 1 are conjugate in J. Each such subgroup is conjugate to J θ for a unique 0 ď θ ď m. (ii) If n is even, then tpΛq P ttpΛ 1 q, n ´tpΛ 1 qu if and only if the associated maximal parahoric subgroups J Λ and J Λ 1 are conjugate in J. Each such subgroup is conjugate to J θ for a unique 0 ď θ ď t m 2 u.

Thus, there are m `1 conjugacy classes of maximal parahoric subgroups when n is odd, and only t m 2 u `1 when n is even. If n is odd the subgroups J θ are pairwise non conjugate, whereas J θ is conjugate to J m´θ when n is even.

Remark. The special maximal compact subgroups are the conjugates of J 0 and of J m . When n is odd, the conjugates of J m are hyperspecial.

Proof. For the first point, assume that tpΛq " tpΛ 1 q. By 3.1.2.1 Proposition, we can find g P J such that g ¨Λ " Λ 1 . Therefore J Λ 1 " J g¨Λ " g J Λ , the two parahoric subgroups are conjugate. For the converse, assume that J Λ 1 " g J Λ for some g P J. Then J Λ 1 " J g¨Λ . By 3.1.3.1 there is some k P Z such that Λ 1 " p k g ¨Λ or pΛ 1 q _ " p k g ¨Λ. This implies that tpΛq " tpΛ 1 q. Indeed, it is clear in the first case, and in the second case we have in particular pΛ 1 q _ P L. Since n is odd, by 3.1.2.2 we have tpΛ 1 q " tppΛ 1 q _ q, so that we are done. For the second point, if tpΛ 1 q " tpΛq then we reason the same way as above. If tpΛ 1 q " n ´tpΛq then Λ 1 and Λ _ have the same type. By the first case, we know that J Λ 1 and J Λ _ " J Λ are conjugate. The converse goes the same way as above, except that the case pΛ 1 q _ " p k g ¨Λ now implies that tpΛ 1 q " n ´tpΛq therefore we are done.

3.1.3.3

As another corollary of 3.1.3.1 we may also describe the normalizers of the maximal parahoric subgroups.

Corollary. Let Λ P L. If tpΛq " n ´tpΛq then the normalizer of J Λ in J is N J pJ Λ q " ZpJqJ Λ . Otherwise, n is even and there exists an element h 0 P J such that h 2 0 " p ¨id and N J pJ λ q is the subgroup generated by J Λ and h 0 . In particular, ZpJqJ Λ is a subgroup of index 2 in N J pJ Λ q.

Remark. The condition tpΛq " n ´tpΛq is automatically satisfied if n is odd. If n is even, it is satisfied when tpΛq " m `1, this is the case in particular when m is odd.

Proof. It is clear that ZpJqJ Λ Ă N J pJ Λ q. Conversely, let g P N J pJ Λ q, so that we have J Λ " g J Λ " J g¨Λ . We apply 3.1.3.1 to deduce the existence of k P Z such that g ¨Λ " p k Λ (case 1) or g ¨Λ " p k Λ _ (case 2). If we are in case 1, then g P p k J Λ Ă ZpJqJ Λ and we are done. If n is even, the assumption that tpΛq " n ´tpΛq makes the case 2 impossible. If n is odd and we are in case 2, then in particular Λ _ P L. By 3.1.2.2, we must have Λ " p i Λ _ for some even i P Z.

In particular, we are also in case 1. Therefore, no matter the parity of n, we are always in case 1. Assume now that tpΛq " n ´tpΛq, in particular n and m are both even. We write m " 2m 1 so that tpΛq " 2m 1 `1 and we solve the case Λ " Λ m 1 first. Recall the element g 0 that was defined in 3.1.1.7. By direct computation, we see that g 0 ¨Λm 1 " pΛ _ m 1 . Therefore g 0 J m 1 " J pΛ _ m 1 " J m 1 so that g 0 P N J pJ m 1 q. Now let g be any element normalizing J m , so that J m 1 " g J m 1 " J g¨Λ m 1 . According to 3.1.3.1 there exists k P Z such that g¨Λ m 1 " p k Λ m 1 or g¨Λ m 1 " p k Λ _ m 1 " p k´1 g 0 ¨Λm 1 . In the first case we have g P p k J m 1 and in the second case we have g P p k´1 g 0 J m 1 . Because g 2 0 " p ¨id, the claim is proved with h 0 " g 0 . In the general case, we have tpΛq " 2m 1 `1 " tpΛ m 1 q. By 3.1.2.1 there exists some g P J such that Λ " g ¨Λm 1 . Then N J pΛq " g N J pΛ m 1 q so that the claim follows with h 0 :" gg 0 g ´1.

3.1.3.4

Let J ˝be the kernel of α : J Ñ Z. In other words, J ˝is the subgroup of J consisting of all g P J whose multiplier cpgq is a unit in Z p . We have an isomorphism J{J ˝» Z induced by α when n is even, and by 1 2 α when n is odd. Note that J ˝contains all the compact subgroups of J, in particular J Λ Ă J ˝for every Λ P L. Let K be the subgroup generated by all the J Λ for Λ P L having maximal orbit type tpΛq " 2m `1. We will prove the following result.

Proposition. We have K " J ˝.

The proof requires the following lemma.

Lemma. Let i P Z such that ni is even and let Λ P L i be a lattice of maximal orbit type. Let Λ 1 , Λ 2 P L i such that Λ 1 X Λ and Λ 2 X Λ belong to L i . There exists g P J Λ such that g ¨Λ1 " Λ 2 if and only if tpΛ 1 q " tpΛ 2 q and tpΛ 1 X Λq " tpΛ 2 X Λq.

Proof. The forward direction is clear because the action of J preserves the types of the lattices. We prove the converse. Since J acts transitively on L while preserving types and inclusions, it is enough to look at the case i " 0 and Λ " Λ m " Λp0, . . . , 0q. Let 0 ď θ ´ď θ `ď m. We fix a certain Λ 1 P L 0 such that tpΛ 1 q " 2θ ``1 and tpΛ 1 X Λq " 2θ ´`1, and we prove that any Λ 2 P L 0 satisfying the hypotheses of the lemma is in the J m -orbit of Λ 1 . We define Λ 1 " Λp0 θ ´, 1 θ `´θ ´, 1 m´θ `, 0, 0 m´θ `, ´1θ `´θ ´, 0 θ ´q where the 0 in the middle stands for 0 when n is odd and the pair p0, 0q when n is even. Then, we have Λ 1 X Λ " Λp0 θ ´, 1 m´θ ´, 0, 0 m´θ ´, 0 θ ´q so that Λ 1 satisfies the required conditions. Let Λ 2 be as in the lemma. Let L be a Witt decomposition of V such that the corresponding apartment A L contains both Λ and Λ 2 . Since J m acts transitively on the set of apartments containing Λ m , we can find some g P J m such that g ¨AL " A e . Up to replacing Λ 2 by g ¨Λ2 , we may then assume that Λ 2 P A e . Therefore, there exists integers r ´m, . . . , r m , s such that Λ 2 " Λpr ´m, . . . , r ´1, s, r 1 , . . . , r m q.

Since Λ 2 P L 0 , by 3.1.2.4 we have s " 0 and r j `r´j P t0, 1u for all 1 ď j ď m. Let us write r ´j " r j `ǫj where ǫ j P t0, 1u. Since tpΛ 2 q " 2θ ``1, there are θ `indices 1 ď j 1 ď . . . ď j θ `ď m such that ǫ j " 0 if and only if j is one of the j k 's. Moreover, we have Λ 2 X Λ " Λ pmaxp´r m `ǫm , 0q, . . . , maxp´r 1 `ǫ1 , 0q, 0, maxpr 1 , 0q, . . . , maxpr m , 0qq .

This lattice is in L 0 , thus for every 1 ď j ď m we have 0 ď maxp´r j `ǫj , 0q `maxpr j , 0q ď 1. Hence, if j " j k for some k then ǫ j " 0 and maxp´r j `ǫj , 0q `maxpr j , 0q " maxp´r j , 0q `maxpr j , 0q " |r j |.

Thus, |r j | " 0 or 1. If j " j k for all k, then ǫ j " 1 and maxp´r j `ǫj , 0q `maxpr j , 0q " maxp´r j `1, 0q `maxpr j , 0q "

1 2 `|r j | `|r j ´1| 2 .
This sum is a positive integer between 0 and 1, therefore it is always 1. It means that |r j | |r j ´1| " 1 and as a consequence, r j " 0 or 1. Lastly, we have tpΛ 2 X Λq " 2θ ´`1 so there are exactly θ ´indices j for which the sum maxp´r j `ǫj , 0q `maxpr j , 0q is zero. As we have just seen, this may only happen when j is one of the j k 's. Thus, among the indices j " j 1 , . . . , j θ `, there are exactly θ ´of them for which pr ´j , r j q " p0, 0q, and for the others we have pr ´j , r j q " p1, ´1q or p´1, 1q. If j is not one of the j k 's, we have pr ´j , r j q " p0, 1q or p1, 0q. In other words, the pairs of indices pr ´j , r j q are, up to shifts and ordering, the same as the corresponding pairs of indices defining Λ 1 . By considering appropriate permutation matrices, we may change a pair pr ´j , r j q into pr j , r ´j q and we may change the order so that Λ 2 is sent to Λ 1 . This transformation defines an element of J which stabilizes Λ " Λp0, . . . , 0q.

We may now prove the proposition.

Proof. It is clear that K Ă Kerpαq, so we prove the reverse inclusion. Let g 0 P J ˝. We will write g 0 as a product of elements in J, each of which fixes some lattice of maximal orbit type in the Bruhat-Tits building. We write Λ :" Λ m " Λp0, . . . , 0q and Λ 0 :" g 0 ¨Λ. Since g 0 P J ˝, we have Λ 0 P L 0 . We would like to send Λ 0 back to Λ by using elements of K only. Let L be some Witt decomposition of V such that the corresponding apartment A L contains both Λ and Λ 0 . We can find some g 1 P J Λ which sends A L to A e . We define g 1 :" g 1 g 0 and Λ 1 :" g 1 ¨Λ. Then Λ 1 P L 0 and it belongs to the apartment A e . Therefore, there exists integers r ´m, . . . , r m , s such that Λ 1 " Λpr ´m, . . . , r ´1, s, r 1 , . . . , r m q.

Since Λ 1 P L 0 and its orbit type is maximal, we have s " 0 and r ´j " ´rj for all 1 ď j ď m. Let 1 ď j 1 ă . . . ă j a ď m be the indices j for which r j is odd. We have 0 ď a ď m. For 1 ď j ď m we write r j " 2r 1 j `1 if j is some of the j 1 k s and r j " 2r 1 j otherwise. We also write r 1 ´j " ´r1 j , so that we have r ´j " 2r 1 ´j ´1 if j is some of the j k 's and r ´j " 2r 1 ´j otherwise. We define g 2 the endomorphism of V sending e ´j to p 2r 1 j e j for ´m ď j ď m and j " 0, and which acts like identity on V an . Then g 2 is an element of J with multiplier equal to 1. Moreover, g 2 stabilizes the lattice Λpr 1 ´m, . . . , r 1 ´1, 0, r 1 1 , . . . , r 1 m q P L 0 whose orbit type is maximal, therefore g 2 P K. We define g 2 :" g 2 g 1 and Λ 2 :" g 2 ¨Λ P L 0 . Concretely, the lattice Λ 2 still lies in the apartment A e and its coefficients are obtained from those of Λ 1 by replacing each pair pr ´jk , r j k q by p1, ´1q and the other pairs pr ´j , r j q by p0, 0q. Let us note that if a " 0 then we already have Λ 2 " Λ. Let us now assume that a ą 0. The intersection of the lattices Λ 2 and Λ has the following shape.

Λ 2 X Λ " Λp 0 or 1, . . . , 0 or 1 l jh n a times 1 and m´a times 0 , 0, 0 m q.

The coefficient takes the value 1 if and only if its index is one of the ´jk 's. This is a lattice in L 0 of orbit type 2pm ´aq `1. We will use 3.1.3.4 Lemma in order to send Λ 2 to Λ while fixing some lattice of maximal orbit type. In order to find this lattice, we need to leave the apartment A e . Let δ P Z p2 such that σpδq " ´δ. We define the following vectors

f j " $ ' &
' % e j if j is not one of the ˘jk 's. pe ´jk if j " ´jk . p ´1e j k `δe ´jk if j " j k .

We also define f an

i " e an i for i P t0, 1u (the case i " 1 only occurs if n is even). All together, these vectors form a basis f of V. We write Λ f for the Z p 2 -lattice generated by the basis f . One may check that xf j , f j 1 y " δ j 1 ,´j for every j and j 1 . It follows that Λ f P L 0 and it has maximal orbit type. It turns out that both intersections Λ 2 X Λ f and Λ X Λ f are equal to Λ 2 X Λ, as we prove in the following two points.

-Λ 2 X Λ f : The lattice Λ 2 XΛ f contains all the vectors e j where j is not of the ˘jk 's. It also contains the vectors pe ´jk and p¨pp ´1e j k `δe ´jk q " e j k `δpe ´jk for all 1 ď k ď a. Therefore, it must contain the vectors e j k 's as well. This gives the inclusion Λ 2 X Λ Ă Λ 2 X Λ f . For

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q the converse, if x P Λ f then we may write

x " ÿ

j "˘j k µ j e j `s ÿ k"1 λ k pe ´jk `λ1 k pp ´1e j k `δe ´jk q " ÿ j "˘j k µ j e j `s ÿ k"1 pλ k p `λ1 k δqe ´jk `λ1 k p ´1e j k
with the scalars µ j , λ k and λ 1 k in Z p 2 . If moreover x P Λ 2 then in the last formula, we must have λ k p `λ1 k δ P pZ p 2 . It follows that the scalars λ 1 k belong to pZ p 2 and thus x P Λ 2 X Λ. -Λ X Λ f : By the same arguments as above, we prove that Λ 2 X Λ Ă Λ X Λ f . For the converse, let x P Λ f as above. If moreover x P Λ then the scalars λ 1 k are elements of pZ p 2 . It implies that λ k p `λ1 k δ P pZ p 2 , whence x P Λ 2 X Λ. Eventually we may apply 3.1.3.4 Lemma to the lattices Λ f , Λ 2 and Λ. It gives the existence of an element g 3 P J which stabilizes Λ f and sends Λ 2 to Λ. We write g 3 :" g 3 g 2 . It follows that g 3 ¨Λ " Λ, therefore g 3 P J Λ Ă K. But g 3 " g 3 g 2 g 1 g 0 and each of the elements g 1 , g 2 and g 3 also lies in K. Therefore g 0 P K as well.

3.1.4

Counting the closed Bruhat-Tits strata 3.1.4.1 In this section we count the number of closed Bruhat-Tits strata which contain or which are contained in another given one. Let d ě 0 and consider V a d-dimensional F p 2 -vector space equipped with a non degenerate hermitian form. This structure is uniquely determined up to isomorphism as we are working over a finite field. As in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF], for P d 2 T ď r ď d, we define N pr, V q :" tU | U is an r-dimensional subspace of V such that U K Ă U u, νpr, dq :" #N pr, V q, where U K denotes the orthogonal of U with respect to the hermitian form on V . As remarked in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF], the set N pr, V q can be seen as the set of rational points of a certain flag variety for the unitary group of V .

Proposition ([VW11] Corollary 5.7). Let Λ P L. Write tpΛq " 2θ `1 for some 0 ď θ ď m.

-Let θ 1 be an integer such that 0 ď θ 1 ď θ. The number of closed Bruhat-Tits strata of dimension θ 1 contained in M Λ is νpθ `θ1 `1, 2θ `1q. -Let θ 1 be an integer such that θ ď θ 1 ď m. The number of closed Bruhat-Tits strata of dimension θ 1 containing M Λ is νpn ´θ ´θ1 ´1, n ´2θ ´1q.

These follows from 3.1.2.9 Proposition and Remark. Another way to formulate the proposition is to say that νpθ `θ1 `1, 2θ `1q (resp. νpn ´θ ´θ1 ´1, n ´2θ ´1q) is the number of vertices of type 2θ 1 `1 in the Bruhat-Tits building of J which are neighbors of a given vertex of type 2θ `1 for θ 1 ď θ (resp. θ 1 ě θ). `p2r´d`j ´p´1q 2r´d`j śd´r j"1 pp 2j ´1q

Proof. Recall that for any integer k, we denote by A k the k ˆk matrix having 1 in the antidiagonal and 0 everywhere else. We fix a basis pe 1 , . . . , e d q of V in which the hermitian form is represented by the matrix A d . We denote by U 0 the subspace generated by the vectors e 1 , . . . , e r . Then the orthogonal of U 0 is generated by e 1 , . . . , e d´r . Since r is an integer between P d 2 T and d, we have 0 ď d ´r ď r and therefore U 0 contains its orthogonal. Thus, U 0 defines an element of N pr, V q. The unitary group UpV q » U d pF p q acts on the set N pr, V q: an element g P UpV q sends the subspace U to gpU q. This action is transitive. Indeed, any U P N pr, V q can be sent to U 0 by using an equivalent of the Gram-Schmidt orthogonalization process over F p 2 (note that p " 2q. The stabilizer of U 0 in U d pF p q is the standard parabolic subgroup

P 0 :" $ ' & ' % ¨B ˚0 M 0 0 F pBq ‹ 'P U d pF p q ˇˇˇˇˇˇB P GL d´r pF p 2 q, M P U 2r´d pF p q , / .
/ -.

Here, F pBq " A d´r pB ppq q ´T A d´r where B ppq is the matrix B with all coefficients raised to the power p. Therefore, the set N pr, V q is in bijection with the quotient U d pF p q{P 0 . The order of U d pF p q is well known and given by the formula #U d pF p q " p dpd´1q 2 d ź j"1 `pj ´p´1q j ˘.

It remains to compute the order of P 0 . We have a Levi decomposition P 0 " L 0 N 0 with L 0 X N 0 " t1u where

L 0 :" $ ' & ' % ¨B 0 0 0 M 0 0 0 F pBq ‹ 'P U d pF p q ˇˇˇˇˇˇB P GL d´r pF p 2 q, M P U 2r´d pF p q , / . / - , N 0 :" $ ' & ' % ¨1 X Z 0 1 Y 0 0 1 ‹ 'P U d pF p q ˇˇˇˇˇˇX
P M d´r,2r´d pF p 2 q, Y P M 2r´d,d´r pF p 2 q, Z P M d´r pF p 2 q , / .

/ -.

The order of L 0 is given by #L 0 " #GL d´r pF p 2 q#U 2r´d pF p q " p pd´rqpd´r´1q`p 2r´dqp2r´d´1q 2 d´r ź j"1 `p2j ´1˘2 r´d ź j"1 `pj ´p´1q j ˘.

As for N 0 , we need some more conditions on the matrices X, Y and Z. By direct computations, one checks that such a matrix belongs to U d pF p q if and only if Y " ´A2r´d pX ppq q T A d´r , Z `Ad´r pZ ppq q T A d´r " XY P M d´r pF p 2 q.

Thus, X is any matrix of size pd ´rq ˆp2d ´rq and Y is determined by X. Let us look at the second equation. The matrix A d´r pZ ppq q T A d´r is the reflexion of Z ppq with respect to the antidiagonal. The equation implies that the coefficients below the antidiagonal of Z determine those above the antidiagonal. Furthermore, if z is a coefficient in the antidiagonal then the equation determines the value of Trpzq " z `zp , where Tr : F p 2 Ñ F p is the trace relative to the extension F p 2 {F p . The trace is surjective and its kernel has order p. Thus, there are only p possibilities for each antidiagonal coefficient. Putting things together, the order of N 0 is given by #N 0 " p 2pd´rqp2r´dq ¨p2 pd´rqpd´r´1q 2 ¨pd´r " p pd´rqp3r´dq

where the three terms take account respectively of the choice of X, the choice of the coefficients below the antidiagonal of Z and the choice of the coefficients in the antidiagonal of Z.

Hence the order of P 0 is given by

#P 0 " #L 0 #N 0 " p dpd´1q 2 d´r ź j"1 `p2j ´1˘2 r´d ź j"1 `pj ´p´1q j ˘.
Upon taking the quotient νpr, dq " #U d pF p q{#P 0 , the result follows.

In particular with r " d ´1, we obtain νpd ´1, dq " pp d´1 ´p´1q d´1 qpp d ´p´1q d q p 2 ´1 .

If d " 2δ is even, it is equal to pp d´1 `1q ř δ´1 j"0 p 2j , and if d " 2δ `1 is odd, it is equal to pp d `1q ř δ´1 j"0 p 2j . This coincides with the formula given in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] Example 5.6.

3.2

The cohomology of a closed Bruhat-Tits stratum 3.2.1 In [START_REF] Muller | Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature (1,n-1)[END_REF], we computed the cohomology groups H ' c pM Λ bF, Q ℓ q of the closed Bruhat-Tits strata (recall that F denotes an algebraic closure of F p ). The computation relies on the Ekedahl-Oort stratification on M Λ which, in the language of Deligne-Lusztig varieties, translates into a stratification by Coxeter varieties for unitary groups of smaller sizes. The cohomology of Coxeter varieties is well known thanks to the work of Lusztig in [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF]. In order to state our results, we recall the classification of unipotent representations of the finite unitary group over Q ℓ .

3.2.2

Let q be a power of prime number p, and let G be a reductive connected group over an algebraic closure F of F p . Assume that G is equipped with an F q -structure induced by a Frobenius morphism F . Let G " G F be the associated finite group of Lie type. Let pT, Bq be a pair consisting of an F -stable maximal torus T and an F -stable Borel subgroup B containing T. Let W " WpTq denote the Weyl group of G. The Frobenius F induces an action on W. For w P W, let 9 w be a representative of w in the normalizer N G pTq of T. By the Lang-Steinberg theorem, one can find g P G such that 9 w " g ´1F pgq. Then g T :" gTg ´1 is another F -stable maximal torus, and w P W is said to be the type of g T with respect to T. Every F -stable maximal torus arises in this manner. According to [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF] Corollary 1.14, the G-conjugacy class of g T only depends on the F -conjugacy class of w in the Weyl group W. Here, two elements w and w 1 in W are said to be F -conjugates if there exists some element τ P W such that w " τ w 1 F pτ q ´1. For every w P W, we fix T w an F -stable maximal torus of type w with respect to T. The Deligne-Lusztig induction of the trivial representation of T w is the virtual representation of G defined by the formula

R w :" ÿ iě0 p´1q i H i c pX H pwq b F, Q ℓ q,
where X H pwq is the Deligne-Lusztig variety for G given by X H pwq :" tgB P G{B | g ´1F pgq P BwBu.

According to [START_REF] Deligne | Representations of Reductive Groups Over Finite Fields[END_REF] Theorem 1.6, the virtual representation R w only depends on the Fconjugacy class of w in W. An irreducible representation of G is said to be unipotent if it occurs in R w for some w P W. The set of isomorphism classes of unipotent representations of G is usually denoted EpG, 1q following Lusztig's notations.

Remark. Since the center ZpGq acts trivially on the variety X H pwq, any irreducible unipotent representation of G has trivial central character.

3.2.3

Let G and G 1 be two reductive connected group over F both equipped with an F qstructure. We denote by F and F 1 the respective Frobenius morphisms. Let f : G Ñ G 1 be an F q -isotypy, that is a homomorphism defined over F q whose kernel is contained in the center of G and whose image contains the derived subgroup of G 1 . Then, according to [DM14] Proposition 11.3.8, we have an equality EpG, 1q " tρ ˝f | ρ P EpG 1 , 1qu.

Thus, the irreducible unipotent representations of G and of G 1 can be identified. We will use this observation in the case G " U k pF q q and G 1 " GU k pF q q. The corresponding reductive groups are G " GL k and G 1 " GL k ˆGL 1 . The Frobenius morphisms can be defined as F pM q " 9 w 0 pM pqq q ´T 9 w 0 , F 1 pM, cq " pc q 9 w 0 pM pqq q ´T 9 w 0 , c q q.

Here, 9 w 0 is the k ˆk matrix with only 1's in the antidiagonal and M pqq is the matrix M whose entries are all raised to the power q. The isotypy f : G Ñ G 1 is defined by f pM q " pM, 1q. It satisfies F 1 ˝f " f ˝F , it is injective and its image contains the derived subgroup SL n ˆt1u Ă G 1 . Hence, we obtain the following result.

Proposition. The irreducible unipotent representations of the finite groups of Lie type U k pF q q and GU k pF q q can be naturally identified.

Assume that the

Theorem ([LS77] Theorem 2.2).

There is a bijection between | W F and the set of isomorphism classes of irreducible unipotent representations of G.

We recall how the bijection is constructed. According to loc. cit. if V P | W F there is a unique automorphism r F of V of finite order such that

RpV q :" 1 |W| ÿ wPW Tracepw ˝r F | V qR w
is an irreducible representation of G. Then the map V Þ Ñ RpV q is the desired bijection. In the case of U k pF q q or GU k pF q q, the Weyl group W is identified with the symmetric group S k and we have an equality | W F " | W. Moreover, the automorphism r F is the multiplication by w 0 , where w 0 is the element of maximal length in W. Thus, in both cases the irreducible unipotent representations of G are classified by the irreducible representations of the Weyl group W » S k , which in turn are classified by partitions of k or equivalently by Young diagrams, as we briefly recall in the next paragraph.

3.2.5

A partition of k is a tuple λ " pλ 1 ě . . . ě λ r q with r ě 1 and each λ i is a positive integer, such that λ 1 `. . . `λr " k. The integer k is called the length of the partition, and it is denoted by |λ|. A Young diagram of size k is a top left justified collection of k boxes, arranged in rows and columns. There is a correspondance between Young diagrams of size k and partitions of k, by associating to a partition λ " pλ 1 , . . . , λ r q the Young diagram having r rows consisting successively of λ 1 , . . . , λ r boxes. We will often identify a partition with its Young diagram, and conversely. For example, the Young diagram associated to λ " p3 2 , 2 2 , 1q is the following one.

To any partition λ of k, one can naturally associate an irreducible character χ λ of the symmetric group S k . An explicit construction is given, for instance, by the notion of Specht modules as explained in [Jam84] 7.1. We will not recall their definition.

3.2.6

The irreducible unipotent representation of U k pF q q (resp. GU k pF q q) associated to χ λ by the bijection of 3.2.4 is denoted by ρ U λ (resp. ρ GU λ ). In virtue of 3.2.3, for every λ we have ρ U λ " ρ GU λ ˝f , where f : U k pF q q Ñ GU k pF q q is the inclusion. Thus, it is harmless to identify ρ U λ and ρ GU λ so that from now on, we will omit the superscript. The partition pkq

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q corresponds to the trivial representation and p1 k q to the Steinberg representation. The degree of the representations ρ λ is given by expressions known as hook formula. Given a box l in the Young diagram of λ, its hook length hp l q is 1 plus the number of boxes lying below it or on its right. For instance, in the following figure the hook length of every box of the Young diagram of λ " p32 , 2 2 , 1q has been written inside it. Proposition ([GP00] Propositions 4.3.5). Let λ " pλ 1 ě . . . ě λ r q be a partition of n. The degree of the irreducible unipotent representation ρ λ is given by the following formula degpρ λ q " q apλq ś k i"1 q i ´p´1q i ś lPλ q hplq ´p´1q hplq where apλq " ř r i"1 pi ´1qλ i .

3.2.7

We may describe the cuspidal support of the unipotent representations ρ λ . According to [START_REF] Lusztig | Irreducible representations of finite classical groups[END_REF] 9.2 and 9.4 there exists an irreducible unipotent cuspidal character of U k pF q q (or GU k pF q q) if and only if k is an integer of the form k " tpt`1q 2 for some t ě 0, and when that is the case it is the one associated to the partition ∆ t :" pt, t´1, . . . , 1q, whose Young diagram has the distinctive shape of a reversed staircase. Here, as a convention U 0 pF q q " GU 0 pF q q denotes the trivial group. For example, here are the Young diagrams of ∆ 1 , ∆ 2 and ∆ 3 . Of course, the one of ∆ 0 the empty diagram.

We consider an integer t ě 0 such that k decomposes as k " 2e `tpt`1q 2 for some e ě 0. Let G denote U k pF q q or GU k pF q q, and consider L t the subgroup consisting of block-diagonal matrices having one middle block of size tpt`1q 2 and all other blocks of size 1. This is a standard Levi subgroup of G. For U k pF q q, it is isomorphic to GL 1 pF q 2 q e ˆU tpt`1q 2 pF q q whereas in the case of GU k pF q q it is isomorphic to G ´U1 pF q q e ˆU tpt`1q 2 pF q q ¯. In both cases, L t admits a quotient which is isomorphic to a group of the same type as G but of size tpt`1q 2 . We write ρ t for the inflation to L t of the unipotent cuspidal representation ρ ∆t of this quotient. If λ is a partition of k, the cuspidal support of the representation ρ λ is given by exactly one of the pair pL t , ρ t q up to conjugacy, where t ě 0 is an integer such that for some e ě 0 we have k " 2e `tpt`1q 2 . Note that in particular k and tpt`1q 3.2.8 Given an irreducible unipotent representation ρ λ , there is a combinatorical way to determine the Harish-Chandra series to which it belongs, as we recalled in [START_REF] Muller | Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature (1,n-1)[END_REF] Section 2. We consider the Young diagram of λ. We call domino any pair of adjacent boxes in the diagram. It may be either vertical or horizontal. We remove dominoes from the diagram of λ so that the resulting shape is again a Young diagram, until one can not proceed further. This process results in the Young diagram of the partition ∆ t for some t ě 0, and it is called the 2-core of λ. It does not depend on the successive choices for the dominoes. Then, the representation ρ λ has cuspidal support pL t , ρ t q if and only if λ has 2-core ∆ t . For instance, the diagram λ " p3 2 , 2 2 , 1q given in 3.2.5 has 2-core ∆ 1 , as it can be determined by the following steps. We put crosses inside the successive dominoes that we remove from the diagram. Thus, the unipotent representation ρ λ of U 11 pF q q or GU 11 pF q q has cuspidal support pL 1 , ρ 1 q, so in particular it is a principal series representation.

ˆùñ ˆˆùñ ˆùñ ˆˆùñ ˆùñ 3.2.9 From now on, we take q " p. We consider the ℓ-adic cohomology with compact support of a closed Bruhat-Tits stratum M Λ b F, where ℓ is a prime number different from p and Λ P L has orbit type tpΛq " 2θ `1, 0 ď θ ď m. Recall from 3.1.2.10 that the stratum M Λ is equipped with an action of the finite group of Lie type GUpV 0 Λ q » GU 2θ`1 pF p q, and as such it is isomorphic to a Deligne-Lusztig variety. Let F be the Frobenius morphism of GU 2θ`1 pF p q as defined in 3.2.3. Then F 2 induces a geometric Frobenius morphism M Λ b F Ñ M Λ b F relative to the F p 2 -structure of M Λ . Because it is a finite morphism, it induces a linear endomorphism on the cohomology groups, and it is in fact an automorphism. In [START_REF] Muller | Cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature (1,n-1)[END_REF], we computed these cohomology groups in terms of a GU 2θ`1 pF p q ˆxF 2 y-representation.

Theorem. Let Λ P L and write tpΛq " 2θ `1 for some 0 ď θ ď m.

(1) The cohomology group H j c pM Λ bF, Q ℓ q is zero unless 0 ď j ď 2θ. There is an isomorphism

H j c pM Λ b F, Q ℓ q » H 2θ´j c pM Λ b F, Q ℓ q _ pθq
which is equivariant for the action of GU 2θ`1 pF p q ˆxF 2 y.

(2) The Frobenius F 2 acts like multiplication by p´pq j on H j c pM Λ b F, Q ℓ q.

(3) For 0 ď j ď θ we have

H 2j c pM Λ b F, Q ℓ q " minpj,θ´jq à s"0 ρ p2θ`1´2s,2sq .
For 0 ď j ď θ ´1 we have

H 2j`1 c pM Λ b F, Q ℓ q " minpj,θ´1´jq à s"0 ρ p2θ´2s,2s`1q .
Thus, in the cohomology of M Λ all the representations associated to a Young diagram with at most 2 rows occur, and there is no other. Such a diagram has the following general shape.

. . . . . .

Remarks.

Let us make a few comments.

-Part p1q of the theorem follows from general theory of etale cohomology given that the variety M Λ is smooth and projective over F p 2 . The identity is a consequence of Poincaré duality. The notation pθq is a Tate twist, it modifies the action of F 2 by multiplying it with p 2θ . -The cohomology groups of index 0 and 2θ are the trivial representation of GU 2θ`1 pF p q.

-All irreducible representations in the cohomology groups of even index belong to the unipotent principal series, whereas all the ones in the groups of odd index have cuspidal support pL 2 , ρ 2 q. -The cohomology group H j c pM Λ bF, Q ℓ q contains no cuspidal representation of GU 2θ`1 pF p q unless θ " j " 0 or θ " j " 1. If θ " 0 then H 0 c is the trivial representation of GU 1 pF p q " F p2 , and if θ " 1 then H 1 c is the representation ρ ∆ 2 of GU 3 pF p q. Both of them are cuspidal.

Shimura variety and p-adic uniformization of the basic stratum

3.3.1 In this section, we introduce the PEL unitary Shimura variety with signature p1, n ´1q as in [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 6.1 and 6.2, and we recall the p-adic uniformization theorem of its basic (or supersingular) locus. The Shimura variety can be defined as a moduli problem classifying abelian varieties with additional structures, as follows. Let E be a quadratic imaginary extension of Q in which p is inert. Let B{F be a simple central algebra of degree d ě 1 which splits over p and at infinity. Let ˚be a positive involution of the second kind on B, and let V be a non-zero finitely generated left B-module equipped with a non-degenerate ˚-alternating form x¨, ¨y taking values in Q. Assume also that dim E pVq " nd. Let G be the connected reductive group over Q whose points over a Q-algebra R are given by GpRq :" tg P GL EbR pV b Rq | Dc P R ˆsuch that for all v, w P V b R, xgv, gwy " cxv, wyu.

We denote by c : G Ñ G m the multiplier character. The base change G R is isomorphic to a group of unitary similitudes GUpr, sq of a hermitian space with signature pr, sq where r `s " n. We assume that r " 1 and s " n ´1. We consider a Shimura datum of the form pG, Xq, where X denotes the unique GpRq-conjugacy class of homorphisms h : C ˆÑ G R such that for all z P C ˆwe have xhpzq¨, ¨y " x¨, hpzq¨y, and such that the R-pairing x¨, hpiq¨y is positive definite. Such a homomorphism h induces a decomposition

V b C " V 1 ' V 2 . Concretely, V 1 (resp. V 2 )
is the subspace where hpzq acts like z (resp. like z). The reflex field associated to this PEL data, that is the field of definition of V 1 as a complex representation of B, is E unless n " 2 in which case it is Q. Nonetheless, for simplicity we will consider the associated Shimura varieties over E even in the case n " 2.

Remark. As remarked in [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] Section 6, the group G satisfies the Hasse principle, ie. ker 1 pQ, Gq is a singleton. Therefore, the Shimura variety associated to the Shimura datum pG, Xq coincides with the moduli space of abelian varieties that we are going to define.

3.3.2

Let A f denote the ring of finite adèles over Q and let K Ă GpA f q be an open compact subgroup. We define a functor Sh K by associating to an E-scheme S the set of isomorphism classes of tuples pA, λ, ι, ηq where -A is an abelian scheme over S.

-λ : A Ñ p A is a polarization. -ι : B Ñ EndpAq b Q is a morphism
of algebras such that ιpb ˚q " ιpbq : where ¨: denotes the Rosati involution associated to λ, and such that the Kottwitz determinant condition is satisfied:

@b P B, detpιpbqq " detpb | V 1 q. -η is a K-level structure, that is a K-orbit of isomorphisms of BbA f -modules H 1 pA, A f q " Ý Ñ V b A f that is compatible with the other data.
The Kottwitz condition in the third point is independent on the choice of h P X. If K is sufficiently small, this moduli problem is represented by a smooth quasi-projective scheme Sh K over E. When the level K varies, the Shimura varieties form a projective system pSh K q K equipped with an action of GpA f q by Hecke correspondences.

3.3.3

We assume the existence of a Z ppq -order O B in B, stable under the involution ˚, such that its p-adic completion is a maximal order in B Qp . We also assume that there is a Z p -lattice Γ in V b Q p , invariant under O B and self-dual for x¨, ¨y. We may fix isomorphisms

E p » Q p 2 and B Qp » M d pQ p 2 q such that O B b Z p is identified with M d pZ p 2 q.
As a consequence of the existence of Γ, the group G Qp is unramified. Let K 0 :" FixpΓq be the subgroup of GpQ p q consisting of all g such that g ¨Γ " Γ. It is a hyperspecial maximal compact subgroup of GpQ p q. We will consider levels of the form K " K 0 K p where K p is an open compact subgroup of GpA p f q. Note that K is sufficiently small as soon as K p is sufficiently small. By the work of Kottwitz in [kottwitzpoints], the Shimura varieties Sh K 0 K p admit integral models over O E,ppq which have the following moduli interpretation. We define a functor S K p by associating to an O E,ppq -scheme S the set of isomorphism classes of tuples pA, λ, ι, η p q where -A is an abelian scheme over S.

λ : A Ñ p A is a polarization whose order is prime to p. ι : O B Ñ EndpAq b Z ppq is a morphism of algebras such that ιpb ˚q " ιpbq : where ¨: denotes the Rosati involution associated to λ, and such that the Kottwitz determinant condition is satisfied:

@b P O B , detpιpbqq " detpb | V 1 q.
Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

-η p is a K p -level structure, that is a K p -orbit of isomorphisms of BbA p f -modules H 1 pA, A p f q " Ý Ñ V b A p
f that is compatible with the other data. If K p is sufficiently small, this moduli problem is also representable by a smooth quasi-projective scheme over O E,ppq . When the level K p varies, these integral Shimura varieties form a projective system pS K p q K p equipped with an action of GpA p f q by Hecke correspondences. We have a family of isomorphisms

Sh K 0 K p » S K p b O E,ppq E
which are compatible as the level K p varies.

Notation. Unless explicitly mentioned, from now on the notation S K p will refer to the smooth quasi-projective Z p 2 -scheme S K p b O E,ppq Z p 2 . Here, we implicitly use the identification of

E p with Q p 2 .
Therefore, with this convention we have isomorphisms Sh

K 0 K p b E Q p 2 » S K p b Z p 2 Q p 2 compatible
as the level K p varies.

3.3.4

Let S K p :" S K p b Z p 2 F p 2 denote the special fiber of the Shimura variety. It is a smooth quasi-projective variety over F p 2 . Its geometry can be described in terms of the Newton stratification as follows. Recall the Shimura datum introduced in 3.3.1. To any homomorphism h P X, we can associate the cocharacter

µ h : C ˆÑ G C " ğ GalpC{Rq G R
which is given by h : C ˆÑ G R into the summand corresponding to the identity in GalpC{Rq.

The conjugacy class µ of µ h is well-determined by X. The field of definition of µ is by definition the reflex field of the Shimura datum, that is E when n " 2 and Q otherwise. We fix an algebraic closure Q (resp. Q p ) containing E (resp. Q p 2 ). We also fix an embedding ν : Q ãÑ Q p compatible with the identification E p » Q p 2 . We may then consider the local datum pG Qp , µ Qp q where µ Qp is the conjugacy class of cocharacters Q p ˆÑ G Qp induced by µ and ν. Let BpG Qp q denote the set of σ-conjugacy classes in Gp q Q p q where q Q p :" { W pFq Q is the completion of the maximal unramified extension of Q p . As in [kottwitziso], we may associate the Kottwitz set BpG Qp , µ Qp q Ă BpG Qp q. It is a finite set equipped with a partial order. An element b P BpG Qp q is said to be µ Qp ´admissible when it belongs to BpG Qp , µ Qp q. The set BpG Qp q (resp. BpG Qp , µ Qp q) canonically classifies the isomorphism classes of isocrystals with a G Qpstructure (resp. compatible µ Qp , G Qp -structures). Let A K p denote the universal abelian scheme over S K p , and let A K p denote its reduction modulo p. The associated p-divisible group A K p rp 8 s is denoted by X K p . For any geometric point x P S K p , the p-divisible group pX K p q x is equipped with compatible µ Qp , G Qp -structures therefore it determines an element b x P BpG Qp , µ Qp q. For b P BpG Qp , µ Qp q, the set S K p pbq :" tx P S K p | b x " bu is locally closed in S K p . It is the underlying topological space of a reduced subscheme which we still denote by S K p pbq. They are called the Newton strata of the special fiber of the Shimura variety. For a fixed b, as the level K p varies the strata form a projective tower pS K p pbqq K p equipped with an action of GpA p f q by Hecke correspondences.

In [BW05]

, the combinatorics of the Newton stratification is described in the case of a PEL unitary Shimura variety of signature p1, n ´1q. The set BpG Qp , µ Qp q contains t n 2 u `1 elements b 0 ă b 1 ă . . . ă b t n 2 u and we have

S K p " t n 2 u ğ i"0 S K p pb i q.
The stratification is linear, that is the closure of a stratum S K p pb i q is the union of all the strata S K p pb j q for j ď i. The stratum corresponding to b i has dimension m `i. The element b t n 2 u is µ-ordinary, and the corresponding stratum S K p pb t n 2 u q is called the µ-ordinary locus. It is open and dense in S K p . The unique basic element is b 0 , and the corresponding stratum S K p pb 0 q is called the basic stratum. It coincides with the supersingular locus. It is a closed subscheme of S K p .

3.3.6

The geometry of the basic stratum can be described using the Rapoport-Zink space M in a process called p-adic uniformization, see [START_REF] Rapoport | Period Spaces for "p"-divisible Groups (AM-141)[END_REF] and [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF]. Let x be a geometric point of S K p pb 0 q. Since G satisfies the Hasse principle, according to [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] Proposition 3.1.8 the isogeny class of the triple pA x , λ, ιq, consisting of the abelian variety A x together with its additional structures, does not depend on the choice of x. We define I :" AutpA x , λ, ιq.

It is a reductive group over Q. In fact, since we are considering the basic stratum, according to loc. cit. the group I is the inner form of G such that IpA f q " J ˆGpA p f q and IpRq » GUp0, nq, that is the unique inner form of GpRq which is compact modulo center. In particular, one can think of IpQq as a subgroup both of J and of GpA p f q. Let p p S K p q |b 0 denote the formal completion of S K p along the basic stratum.

Theorem ([RZ96] Theorem 6.24).

There is an isomorphism of formal schemes over SpfpZ p 2 q

Θ K p : IpQqz `M ˆGpA p f q{K p ˘" Ý Ñ p p S K p q |b 0
which is compatible with the GpA p f q-action by Hecke correspondences as the level K p varies.

This isomorphism is known as the p-adic uniformization of the basic stratum. The induced map on the special fiber is an isomorphism pΘ K p q s : IpQqz `Mred ˆGpA p f q{K p ˘" Ý Ñ S K p pb 0 q of schemes over SpecpF p 2 q. We denote by M an (resp. p p S K p q an |b 0 ) the smooth analytic space over Q p 2 associated to the formal scheme M (resp. p p S K p q |b 0 ) by the Berkovich functor as defined in [START_REF] Berkovich | Vanishing cycles for formal schemes. II[END_REF]. Note that both formal schemes are special in the sense of loc. cit. so that we may use Berkovich's constructions. These analytic spaces play the role of the generic fibers of the formal schemes over SpfpZ p 2 q. By [Far04] Théorème 3.2.6, p-adic uniformization induces an isomorphism Θ an K p : IpQqz `Man ˆGpA p f q{K p ˘" Ý Ñ p p S K p q an |b 0 of analytic spaces over Q p 2 . We denote by red the reduction map from the generic fiber to the special fiber. It is an anticontinuous map of topological spaces, which means that the preimage of an open subset is closed and the preimage of a closed subet is open. Then, the uniformization on the generic and special fibers are compatible in the sense that the diagram

IpQqz `Man ˆGpA p f q{K p ˘pp S K p q an |b 0 IpQqz `Mred ˆGpA p f q{K p ˘SK p pb 0 q Θ an K p red red pΘ K p qs
is commutative.

3.3.7

The double coset space IpQqzGpA p f q{K p is finite, so that we may fix a system of representatives g 1 , . . . , g s P GpA p f q. For every 1 ď k ď s, we define Γ k :" IpQq X g k K p g ´1 k , which we see as a discrete subgroup of J that is cocompact modulo the center. The left hand side of the p-adic uniformization theorem is isomorphic to the disjoint union of the various quotients of M (or M red or M an ) by the subgroups Γ k Ă J. In particular for the special fiber, it is an isomorphism

pΘ K p q s : s ğ k"1 Γ k zM red " Ý Ñ S K p pb 0 q.
Let Φ k K p be the composition M red Ñ Γ k zM red Ñ Sh ss C p and let Φ K p be the disjoint union of the Φ k K p . The map Φ K p is surjective onto S K p pb 0 q. According to [VW11] Section 6.4, it is a local isomorphism which can be used in order to transport the Bruhat-Tits stratification from M red to S K p pb 0 q. Recall the notations of 3.1.2.3.

Proposition ([VW11]

Proof of Proposition 6.5). Let Λ P L. For any 1 ď k ď s, the restriction of Φ k K p to M Λ is an isomorphism onto its image.

We will denote by S K p ,Λ,k the scheme theoretic image of M Λ through Φ k . A subscheme of the form S K p ,Λ,k is called a closed Bruhat-Tits stratum of the Shimura variety. Together, they form the Bruhat-Tits stratification of the basic stratum, whose combinatorics is described by the union of the complexes Γ k zL.

3.4

The cohomology of the Rapoport-Zink space at maximal level 3.4.1 The spectral sequence associated to an open cover of M an 3.4.1.1 As in 3.3.6, we consider the generic fiber M an of the Rapoport-Zink space as a smooth Berkovich analytic space over Q p 2 . Let red : M an Ñ M red be the reduction map. If Z is a locally closed subset of M red , then the preimage red ´1pZ q is called the analytical tube over Z. It is an analytic domain in M an and it coincides with the generic fiber of the formal completion of M red along Z. If i P Z such that ni is even, then the tube red ´1pM i q " M an i is open and closed in M an and we have

M an " ğ niP2Z M an i .
If Λ P L, we define U Λ :" red ´1pM Λ q the tube over M Λ . The action of J on M induces an action on the generic fiber M an such that red is J-equivariant. By restriction it induces an action of J Λ on U Λ . The analytic space M an and each of the open subspaces U Λ have dimension n ´1.

3.4.1.2

We fix a prime number ℓ " p. In [START_REF] Berkovich | Étale cohomology for non-Archimedean analytic spaces[END_REF], Berkovich developped a theory of étale cohomology for his analytic spaces. Using it we may define the cohomology of the Rapoport-Zink space M an by the formula

H ' c pM an p b C p , Q ℓ q :" lim Ý Ñ U H ' c pU p b C p , Q ℓ q " lim Ý Ñ U lim Ð Ý n H ' c pU p b C p , Z{ℓ n Zq b Q ℓ
where U goes over all relatively compact open of M an . These cohomology groups are equipped with commuting actions of J and of W , the Weyl group of Q p 2 . The J-action causes no problem of interpretation, but the W -action needs explanations. Let τ :" σ 2 be the Frobenius relative to F p 2 . We fix a lift Frob P W of the geometric Frobenius τ ´1 P GalpF{F p 2 q. The inertia subgroup I Ă W acts on H ' c pM an p b C p , Q ℓ q via the coefficients C p , whereas Frob acts via the Weil descent datum defined by Rapoport and Zink in [START_REF] Rapoport | Period Spaces for "p"-divisible Groups (AM-141)[END_REF] 3.48, as we explain now. Recall the standard unitary p-divisible group X introduced in 3.1.1.1. Let

F X : X b F Ñ τ ˚pX b Fq
denote the Frobenius morphism relative to F p 2 . Let pM p b O q Qp q τ be the functor defined by pM p b O q Qp q τ pSq :" MpS τ q for all O q Qp -scheme S where p is locally nilpotent. Here, S τ denotes the scheme S but with structure morphism the composition S Ñ SpecpO q Qp q τ Ý Ñ SpecpO q Qp q. The Weil descent datum is the isomorphism

α RZ : M p b O q Qp " Ý Ñ pM p b O q
Qp q τ given by pX, ι, λ, ρq P MpSq Þ Ñ pX, ι, λ, F X ˝ρq. We may describe this in terms of k-rational points, where k is a perfect field extension of F. Since we use covariant Dieudonné theory, the relative Frobenius F X corresponds to the Verschiebung V 2 in the Dieudonné module. By construction of X, we have V 2 " pτ ´1. Therefore, if S " Specpkq with k{F p 2 perfect, then α RZ sends a Dieudonné module M P Mpkq to pτ ´1pM q. Since Frob P W is a geometric Frobenius element, its action on the cohomology of M an is induced by the inverse α ´1 RZ .

Remark. The Rapoport-Zink space is defined over Z p 2 and this rational structure is induced by the effective descent datum pα ´1 RZ , with p " p ¨id seen as an element of the center of J. It sends a point M to τ pM q. Consequently, in the following we will write τ :" pp ´1 ¨id, Frobq P J ˆW , and we refer to it as the rational Frobenius. We note that p ´1 ¨id comes from contravariance of cohomology with compact support: the action of g P J on the cohomology of M an is induced by the action of g ´1 on the space M an . Notation. In order to shorten the notations, we will omit the coefficients C p . Thefore we write H ' c pM an , Q ℓ q and similarly for subspaces of M an .

3.4.1.3

The cohomology groups H ' c pM an , Q ℓ q are concentrated in degrees 0 to 2 dimpM an q " 2pn ´1q. According to [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] Corollaire 4.4.7, these groups are smooth for the J-action and continous for the I-action. In a similar way as for M an , we can also define the cohomology groups H ' c pM an i , Q ℓ q for every i P Z such that ni is even. The action of an element g P J induces an isomorphism g :

H ' c pM an i , Q ℓ q " Ý Ñ H ' c pM an i`αpgq , Q ℓ q.
In particular, the action of Frob gives an isomorphism from the cohomology of M an i to that of M an i`2 . Let pJ ˆW q ˝be the subgroup of J ˆW consisting of all elements of the form pg, uFrob j q with u P I and αpgq " ´2j. In fact, we have pJ ˆW q ˝" pJ ˝ˆI qτ Z where J ˝Ă J is the subgroup introduced in 3.1.3.4. Each group H ' c pM an i , Q ℓ q is a pJ ˆW q ˝-representation, and we have an isomorphism

H ' c pM an , Q ℓ q » c ´Ind JˆW pJˆW q ˝H' c pM an 0 , Q ℓ q.
In particular, when H k c pM an , Q ℓ q is non-zero it is infinite dimensional. However, by loc. cit. Proposition 4.4.13, these cohomology groups are always of finite type as J-modules.

3.4.1.4

In order to obtain information on the cohomology of M an , we study the spectral sequence associated to the covering by the open subspaces U Λ for Λ P L. The spaces U Λ satisfy the same incidence relations as the M Λ , as described in 3.1.2.11 Theorem (1), (2) and (3). As a consequence, the open covering of M an by the tU Λ u is locally finite. For i P Z such that ni is even and for 0 ď θ ď m, we denote by L pθq i the subset of L i whose elements are those lattices of orbit type 2θ `1. We also write L pθq for the union of the L pθq i . Then tU Λ u ΛPL pmq is an open cover of M an . We may apply [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] Proposition 4.2.2 to deduce the existence of the following Čech spectral sequence computing the cohomology of the Rapoport-Zink space, concentrated in degrees a ď 0 and 0 ď b ď 2pn ´1q, E a,b 1 :

à γPI ´a`1 H b c pU pγq, Q ℓ q ùñ H a`b c pM an , Q ℓ q.
Here, for s ě 1 the set I s is defined by

I s :" # γ " pΛ 1 , . . . , Λ s q ˇˇˇˇ@
1 ď j ď s, Λ j P L pmq and U pγq :"

s č j"1 U Λ j " H + .
Necessarily, if γ " pΛ 1 , . . . , Λ s q P I s then there exists a unique i such that ni is even and Λ j P L pmq i for all j. We then define Λpγq :"

s č j"1 Λ j P L i ,
so that U pγq " U Λpγq . In particular, the open subspace U pγq depends only on the intersection Λpγq of the elements in the s-tuple γ.

For s ě 2 and γ " pΛ 1 , . . . , Λ s q P I s , define γ j :" pΛ 1 , . . . , x Λ j , . . . , Λ s q P I s´1 for the ps ´1q-tuple obtained from γ by removing the j-th term. Besides, for Λ, Λ 1 P L i with Λ 1 Ă Λ, we write

f b Λ 1 ,Λ for the natural map H b c pU Λ 1 , Q ℓ q Ñ H b c pU Λ , Q ℓ q induced by the inclusion U Λ 1 Ă U Λ . For a ď ´1, the differential E a,b 1 Ñ E a`1,b 1 is denoted by ϕ b ´a.
It is the direct sum over all γ P I ´a`1 of the maps

H b c pU pγq, Q ℓ q Ñ à δPtγ 1 ,...γ ´a`1 u H b c pU pδq, Q ℓ q v Þ Ñ ´a`1 ÿ j"1 γ j ¨p´1q j`1 f b Λpγq,Λpγ j q pvq.
Here, the notation γ j ¨p´1q j`1 f b Λpγq,Λpγ j q pvq means the vector p´1q j`1 f b Λpγq,Λpγ j q pvq considered inside the summand H b c pU pδq, Q ℓ q corresponding to δ " γ j . We observe that we may have Λpγ j q " Λpγ j 1 q even though γ j " γ j 1 . In such a case, the vectors f b Λpγq,Λpγ j q pvq and f b Λpγq,Λpγ j 1 q pvq are equal in H b c pU pγ j q, Q ℓ q " H b c pU pγ j 1 q, Q ℓ q, but they contribute to two distinct summands in the codomain, namely associated to δ " γ j and δ " γ j 1 . An element g P J acts on the set I s by sending γ to g ¨γ :" pgΛ 1 , . . . , gΛ s q. The action of g

´1 induces an isomorphism H b c pU pγq, Q ℓ q " Ý Ñ H b c pU pg ¨γq, Q ℓ q.
This defines a natural J-action on the terms E a,b 1 , with respect to which the spectral sequence is equivariant.

Remark. The map pα ´1

RZ defines a Weil descent datum on M Λ bF which is effective, and coincides with the natural F p 2 -structure. Hence, the same holds for the analytical tube U Λ p b C p . The descent datum pα ´1 RZ induces the action of τ on the cohomology of U Λ . If γ P I ´a`1 then p ¨γ P I ´a`1 . It follows that each term E a,b 1 is equipped with an action of W . The spectral sequence E is in fact J ˆW -equivariant.

3.4.1.5 First we relate the cohomology of a tube U Λ to the cohomology of the corresponding closed Bruhat-Tits stratum M Λ . We observe that H ' c pU Λ , Q ℓ q is naturally a representation of the subgroup pJ Λ ˆIqτ Z Ă J ˆW .

Proposition. Let Λ P L and let 0 ď b ď 2pn ´1q. There is a pJ Λ ˆIqτ Z -equivariant isomorphism

H b pM Λ b F, Q ℓ q " Ý Ñ H b pU Λ , Q ℓ q
Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

where, on the left-hand side, the inertia I acts trivially and τ acts like the geometric Frobenius F 2 .

In particular, the inertia acts trivially on the cohomology of U Λ .

Proof. Recall the notations of 3.3.7 regarding the Bruhat-Tits stratification on the Shimura variety S K p , where K p is any open compact subgroup of GpA p f q that is small enough. Fix an integer 1 ď k ď s and consider the closed Bruhat-Tits stratum S K p ,Λ,k , that is the isomorphic image of M Λ through Φ k K p . Let Sh K p ,Λ,k be the analytic tube of S K p ,Λ,k inside p p S K p q an |b 0 . By compatibility of the p-adic uniformization, the tube Sh K p ,Λ,k is the isomorphic image of U Λ through pΦ k K p q an , which is the composition M an Ñ Γ k zM an Ñ p p S K p q an |b 0 . Thus, the following diagram is commutative.

U Λ Sh K p ,Λ,k M Λ S K p ,Λ,k " red red "
Berkovich's comparison theorem gives the desired isomorphism. More precisely, let p S K p denote the formal completion of the Shimura variety S K p along its special fiber. Since it is a smooth formal scheme over SpfpZ p 2 q, we may apply [START_REF] Berkovich | Vanishing cycles for formal schemes. II[END_REF] Corollary 3.7 to deduce the existence of a natural isomorphism

H b pS K p ,Λ,k b F, Q ℓ q " Ý Ñ H b pSh K p ,Λ,k , Q ℓ q.
This isomorphism is equivariant for the action of pJ Λ ˆIqτ Z , with the rational Frobenius τ on the right-hand side corresponding to F 2 on the left-hand side.

Remark. It is a priori not possible to use Berkovich's result directly on the Rapoport-Zink space because M is not a smooth formal scheme over SpfpZ 2 p q. In fact, it is not adic unless n " 1 or 2, see [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] Remarque 2.3.5. It is the reason why we have to introduce the Shimura variety in the proof.

Corollary. Let Λ P L and let 0 ď b ď 2pn ´1q. There is a pJ Λ ˆIqτ Z -equivariant isomorphism

H b c pU Λ , Q ℓ q " Ý Ñ H b´2pn´1´θq c pM Λ b F, Q ℓ qpn ´1 ´θq
where tpΛq " 2θ `1.

Proof. This is a consequence of algebraic and analytic Poincaré duality, respectively for U Λ and for M Λ . Indeed, we have

H b c pU Λ , Q ℓ q » H 2pn´1q´b pU Λ , Q ℓ q _ pn ´1q » H 2pn´1q´b pM Λ b F, Q ℓ q _ pn ´1q » H b´2pn´1´θq c pM Λ b F, Q ℓ qpn ´1 ´θq.
3.4.1.6 Let Λ P L and write tpΛq " 2θ `1. If λ is a partition of 2θ `1, recall the unipotent irreducible representation ρ λ of GUpV 0 Λ q » GU 2θ`1 pF p q that we introduced in 3.2.6. It can be inflated to the maximal reductive quotient J Λ » GpUpV 0 Λ q ˆUpV 1 Λ qq, and then to the maximal parahoric subgroup J Λ . With an abuse of notation, we still denote this inflated representation by ρ λ . In virtue of 3.2.9, the isomorphism in the last paragraph translates into the following result.

Proposition. Let Λ P L and write tpΛq " 2θ `1. The following statements hold.

(1) The cohomology group H b c pU Λ , Q ℓ q is zero unless 2pn ´1 ´θq ď b ď 2pn ´1q.

(2) The action of J Λ on the cohomology factors through an action of the finite group of Lie type GUpV 0 Λ q. The rational Frobenius τ acts like multiplication by p´pq b on H b c pU Λ , Q ℓ q.

(3) For 0 ď b ď θ we have

H 2b`2pn´1´θq c pU Λ , Q ℓ q " minpj,θ´jq à s"0 ρ p2θ`1´2s,2sq .
For 0 ď b ď θ ´1 we have

H 2b`1`2pn´1´θq c pU Λ , Q ℓ q " minpj,θ´1´jq à s"0
ρ p2δ´2s,2s`1q .

3.4.1.7

The description of the rational Frobenius action yields the following result.

Corollary. The spectral sequence degenerates on the second page E 2 . For 0 ď b ď 2pn ´1q, the induced filtration on H b c pM an , Q ℓ q splits, ie. we have an isomorphism

H b c pM an , Q ℓ q » à bďb 1 ď2pn´1q E b´b 1 ,b 1 2 .
The action of W on H b c pM an , Q ℓ q is trivial on the inertia subgroup and the action of the rational Frobenius element τ is semisimple. The subspace E b´b 1 ,b 1 2 is identified with the eigenspace of τ associated to the eigenvalue p´pq b 1 .

Remark. In the previous statement, the terms E b´b 1 ,b 1 2 may be zero.

Proof. The pa, bq-term in the first page of the spectral sequence is the direct sum of the cohomology groups H b c pU pγq, Q ℓ q for all γ P I ´a`1 . On each of these cohomology groups, the rational Frobenius τ acts like multiplication by p´pq b . This action is in particular independant of γ and of a. Thus, on the b-th row of the first page of the sequence, the Frobenius acts everywhere as multiplication by p´pq b . Starting from the second page, the differentials in the sequence connect two terms lying in different rows. Since the differentials are equivariant for the τ -action, they must all be zero. Thus, the sequence degenerates on the second page. By the machinery of spectral sequences, there is a filtration on H b c pM an , Q ℓ q whose graded factors are given by the terms E b´b 1 ,b 1 2 of the second page. Only a finite number of these terms are non-zero, and since they all lie on different rows, the Frobenius τ acts like multiplication by a different scalar on each graded factor of the filtration. It follows that the filtration splits, ie. the abutment is the direct sum of the graded pieces of the filtration, as they correspond to the eigenspaces of τ . Consequently, its action is semisimple.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q 3.4.1.8 The spectral sequence E a,b 1 has non-zero terms extending indefinitely in the range a ď 0. For instance, if Λ P L pmq then pΛ, . . . , Λq P I ´a`1 so that E a,b 1 " 0 for all a ď 0 and 2pn ´1 ´mq ď b ď 2pn ´1q. To rectify this, we introduce the alternating Čech spectral sequence. If v P E a,b 1 and γ P I ´a`1 , we denote by v γ P H b c pU pγq, Q ℓ q the component of v in the summand of E a,b 1 indexed by γ. Besides, if γ " pΛ 1 , . . . , Λ ´a`1 q P I ´a`1 and if σ P S ´a`1 then we write σpγq :" pΛ σp1q , . . . , Λ σp´a`1q q P I ´a`1 . For all a, b we define

E a,b
1,alt :" tv P E a,b 1 | @γ P I ´a`1 , @σ P S ´a`1 , v σpγq " sgnpσqv γ u. In particular, if γ " pΛ 1 , . . . , Λ ´a`1 q with Λ j " Λ j 1 for some j "

j 1 then v P E a,b 1,alt ùñ v γ " 0. The subspace E a,b 1,alt Ă E a,b
1 is stable under the action of J ˆW , and the differential

ϕ b ´a : E a,b 1 Ñ E a`1,b 1 sends E a,b
1,alt to E a`1,b 1,alt . Thus, for all b we have a chain complex E ',b 1,alt and the following proposition is well-known.

Proposition ([Sta23] Lemma 01FM). The inclusion map E ',b 1,alt ãÑ E ',b
1 is a homotopy equivalence. In particular we have canonical isomorphisms E a,b 2,alt » E a,b 2 for all a, b.

The advantage of the alternating Čech spectral sequence is that it is concentrated in a finite strip. Indeed, if γ " pΛ 1 , . . . , Λ ´a`1 q P I ´a`1 , let i P Z such that Λpγq P L i . Then all the Λ j 's belong to the set of lattices in L pmq i containing Λpγq. This set is finite of cardinality νpn ´θ ´m ´1, n ´2θ ´1q where tpΛpγqq " 2θ `1 according to 3.1.4.1. Thus, if ´a `1 is big enough then all the γ's in I ´a`1 will have some repetition, so that E a,b 1,alt " 0. Remark. The Lemma 01FM of [START_REF]The Stacks project authors[END_REF] is stated in the context of Čech cohomology of an abelian presheaf F on a topological space X. However, the proof may be adapted to Čech homology of precosheaves such as U Þ Ñ H b c pU, Q ℓ q.

3.4.1.9 For a " 0, we have E 0,b 1,alt " E 0,b 1 by definition. Let us consider the cases b " 2pn ´1 ´mq and b " 2pn ´1 ´mq `1. For such b, it follows from 3.4.1.6 that H b c pU Λ , Q ℓ q " 0 if tpΛq ă t max . If a ď ´1, we have ´a `1 ě 2 so that for all γ " pΛ 1 , . . . , Λ ´a`1 q P I ´a`1 , if there exists j " j 1 such that Λ j " Λ j 1 , then tpΛpγqq ă t max and H b c pU pγq, Q ℓ q " 0. It follows that E a,b 1,alt " 0 for all a ď ´1 and b as above. This observation, along with the previous paragraph, yields the following proposition.

Proposition. We have E 0,2pn´1´mq 2 » E 0,2pn´1´mq 1 . If moreover m ě 1 (ie. n ě 3), then we have E 0,2pn´1´mq`1 2 » E 0,2pn´1´mq`1 1
as well.

3.4.1.10

In order to study the action of J, we may rewrite E a,b 1 conveniently in terms of compactly induced representations. To do this, let us introduce a few more notations. For 0 ď θ ď m and s ě 1, we define I pθq s :" tγ P I s | tpΛpγqq " 2θ `1u. The subset I pθq s Ă I s is stable under the action of J. We denote by N pΛ θ q the finite set N pn ´θ ´m ´1, V 1 θ q as defined in paragraph 3.1.4.1. It corresponds to the set of lattices Λ P L 0 of maximal orbit type tpΛq " 2m `1 containing Λ θ . For s ě 1 we define K pθq s :" tδ " pΛ 1 , . . . , Λ s q | @1 ď j ď s, Λ j P N pΛ θ q and Λpδq " Λ θ u.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q Then K pθq s is a finite subset of I pθq s and it is stable under the action of J θ . If γ P I pθq s , there exists some g P J such that g ¨Λpγq " Λ θ because both lattices share the same orbit type. Moreover, the coset J θ ¨g is uniquely determined, and g ¨γ is an element of K pθq s . This mapping results in a natural bijection between the orbit sets

JzI pθq s " Ý Ñ J θ zK pθq s .
The bijection sends the orbit J ¨α to the orbit J θ ¨pg ¨αq where g is chosen as above. The inverse sends an orbit J θ ¨β to J ¨β. We note that both orbit sets are finite.

We may now rearrange the terms in the spectral sequence.

Proposition. We have an isomorphism

E a,b 1 » m à θ"0 à rδsPJ θ zK pθq ´a`1 c ´Ind J Fixpδq H b c pU Λ θ , Q ℓ q |Fixpδq » m à θ"0 c ´Ind J J θ ´Hb c pU Λ θ , Q ℓ q b Q ℓ rK pθq ´a`1 s ¯,
where Q ℓ rK pθq ´a`1 s is the permutation representation associated to the action of J θ on the finite set K pθq ´a`1 .

Remark. For δ P K pθq s , the group Fixpδq consists of the elements g P J such that g ¨δ " δ. Any such g satisfies gΛpδq " Λpδq, and since Λpδq " Λ θ we have Fixpδq Ă J θ . If δ " pΛ 1 , . . . , Λ s q then Fixpδq is the intersection of the maximal parahoric subgroups J Λ 1 , . . . , J Λ s . We note that in general, Fixpδq is itself not a parahoric subgroup of J since the lattices Λ 1 , . . . , Λ s need not form a simplex in L, as they all share the same orbit type. If however Λ 1 " . . . " Λ s then Fixpδq " J Λ 1 is a conjugate of the maximal parahoric subgroup J m .

Proof. First, by decomposing I ´a`1 as the disjoint union of the I pθq ´a`1 for 0 ď θ ď m, we may write

E a,b 1 " m à θ"0 à γPI pθq ´a`1 H b c pU pγq, Q ℓ q.
For each orbit X P JzI pθq ´a`1 , we fix a representative δ X which lies in K pθq ´a`1 . We may write

E a,b 1 " m à θ"0 à XPJzI pθq ´a`1 à γPX H b c pU pγq, Q ℓ q " m à θ"0 à XPJzI pθq ´a`1 à gPJ{Fixpδ X q g ¨Hb c pU pδ X q, Q ℓ q.
The rightmost sum can be identified with a compact induction from Fixpδ X q to J. Identifying the orbit sets JzI

pθq ´a`1 " Ý Ñ J θ zK pθq ´a`1 , we have E a,b 1 » m à θ"0 à rδsPJ θ zK pθq ´a`1 c ´Ind J Fixpδq H b c pU Λ θ , Q ℓ q |Fixpδq .
By transitivity of compact induction, we have

c ´Ind J Fixpδq H b c pU Λ θ , Q ℓ q |Fixpδq " c ´Ind J J θ c ´Ind J θ Fixpδq H b c pU Λ θ , Q ℓ q |Fixpδq .
Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

Since H b c pU Λ θ , Q ℓ q |Fixpδq is the restriction of a representation of J θ to Fixpδq, applying compact induction from Fixpδq to J θ results in tensoring with the permutation representation of J θ {Fixpδq. Thus

E a,b 1 » m à θ"0 à rδsPJ θ zK pθq ´a`1 c ´Ind J J θ `Hb c pU Λ θ , Q ℓ q b Q ℓ rJ θ {Fixpδqs » m à θ"0 c ´Ind J J θ ¨Hb c pU Λ θ , Q ℓ q b à rδsPJ θ zK pθq ´a`1 Q ℓ rJ θ {Fixpδqs ',
where on the second line we used additivity of compact induction. Now, J θ {Fixpδq is identified with the J θ -orbit J θ ¨δ of δ in K pθq ´a`1 , so that à

rδsPJ θ zK pθq ´a`1 Q ℓ rJ θ {Fixpδqs » Q ℓ r ğ rδsPJ θ zK pθq ´a`1 J θ ¨δs » Q ℓ rK pθq ´a`1 s,
which concludes the proof.

3.4.1.11 By 3.1.2.9, we may identify N pΛ θ q with the set

N pΛ θ q :" tU Ă V 1 θ | dim U " m ´θ and U Ă U K u. Thus, for s ě 1, K pθq s is naturally identified with K pθq s » # δ " pU 1 , . . . , U s q ˇˇˇˇ@ 1 ď j ď s, U j P N pΛ θ q and s č j"1 U j " t0u + .
The action of J θ on K pθq s corresponds to the natural action of GUpV 1 θ q on K pθq s , which factors through an action of the finite projective unitary group PUpV 1 θ q :" UpV 1 θ q{ZpUpV 1 θ qq » GUpV 1 θ q{ZpGUpV 1 θ qq. Thus, the representation Q ℓ rK pθq ´a`1 s of J θ is the inflation, via the maximal reductive quotient as in 3.1.2.8, of the representation Q ℓ rK pθq ´a`1 s of the finite projective unitary group PUpV 1 θ q. When θ " m or when s " 1, we trivially have the following proposition.

Proposition. For s ě 1, we have Q ℓ rK pmq s s " 1. For 0 ď θ ď m ´1, we have Q ℓ rK pθq 1 s " 0. Proof. If δ " pΛ 1 , . . . , Λ s q P K pmq s then Λpδq " Λ m has maximal orbit type t max " 2m `1. For any 1 ď j ď s we have Λ m Ă Λ j , therefore Λ 1 " . . . " Λ s " Λ m . Thus K pmq s is a singleton and so Q ℓ rK pmq s s is trivial. Besides, if θ ă m then K pθq s
is clearly empty. Recall 3.4.1.9 Proposition. We obtain the following corollary.

Corollary. We have

E 0,b 1 » c ´Ind J Jm H b c pU Λm , Q ℓ q.
In particular, we have

E 0,b 2 » $ & % c ´Ind J Jm ρ p2m`1q if b " 2pn ´1 ´mq, c ´Ind J Jm ρ p2m,1q if m ě 1 and b " 2pn ´1 ´mq `1.
Remark. The representation ρ p2m`1q " 1 is the trivial representation of J m .

3.4.1.12 Let us now consider the top row of the spectral sequence, corresponding to b " 2pn ´1q. For Λ 1 Ă Λ, recall the map

f 2pn´1q Λ 1 ,Λ : H 2pn´1q c pU Λ 1 , Q ℓ q Ñ H 2pn´1q c pU Λ , Q ℓ q.
By Poincaré duality, it is the dual map of the restriction morphism H 0 pU Λ , Q ℓ q Ñ H 0 pU Λ 1 , Q ℓ q. Since U Λ is connected for every Λ P L, we have H 0 pU Λ , Q ℓ q » Q ℓ and the restriction maps for Λ 1 Ă Λ are all identity. Thus, E a,2pn´1q 1 is the Q ℓ -vector space generated by I ´a`1 , and the differential ϕ 2pn´1q ´a is given by

γ P I ´a`1 Þ Ñ ´a`1 ÿ j"1 p´1q j`1 γ j .
Using this description, we may compute the highest cohomology group H 2pn´1q c pM an , Q ℓ q explicitely.

Proposition. There is an isomorphism

H 2pn´1q c pM an , Q ℓ q » c ´Ind J J ˝1,
and the rational Frobenius τ acts via multiplication by p 2pn´1q .

Proof. The statement on the Frobenius action is already known by 3.4.1.7 Corollary. Besides, we have H

2pn´1q c pM an , Q ℓ q » E 0,2pn´1q 2 " Cokerpϕ 2pn´1q 1 q. The differential ϕ 2pn´1q 1 is described by pΛ, Λq Þ Ñ 0, @Λ P L pmq , pΛ, Λ 1 q Þ Ñ pΛ 1 q ´pΛq, @Λ, Λ 1 P L pmq such that U Λ X U Λ 1 " H.
Let i P Z such that ni is even, and let Λ, Λ 1 P L pmq i . Since the Bruhat-Tits building BTp r J, Q p q » L i is connected, there exists a sequence Λ " Λ 0 , . . . , Λ d " Λ 1 of lattices in L i such that for all 0 ď j ď d ´1, tΛ j , Λ j`1 u is an edge in L i . Assume that d ě 0 is minimal satisfying this property. Since tpΛq " tpΛ 1 q " t max , the integer d is even and we may assume that tpΛ j q is equal to t max when j is even, and equal to 1 when j is odd. In particular, for all 0

ď j ď d 2 ´1 we have Λ 2j , Λ 2j`2 P L pmq i and U Λ 2j X U Λ 2j`2 " H. Consider the vector w :" d 2 ´1 ÿ j"0 pΛ 2j , Λ 2j`2 q P E ´1,2pn´1q 1 .
Then we compute ϕ 2pn´1q 1 pwq " pΛ 1 q ´pΛq. Thus, Cokerpϕ 2pn´1q 1 q consists of one copy of Q ℓ for each i P Z such that ni is even. Considering the action of J as well, it readily follows that Cokerpϕ

2pn´1q 1 q » c ´Ind J J ˝1.
Remark. The cohomology group H 2pn´1q c pM an , Q ℓ q can also be computed in another way which does not require the spectral sequence. Indeed, we have an isomorphism

H 2pn´1q c pM an , Q ℓ q » c ´Ind J J ˝H2pn´1q c pM an 0 , Q ℓ q.
By definition, we have

H 2pn´1q c pM an 0 , Q ℓ q " lim Ý Ñ U H 2pn´1q c pU p b C p , Q ℓ q,
where U runs over the relatively compact open subspaces of M an 0 . Since U is smooth, Poincaré duality gives

H 2pn´1q c pU p b C p , Q ℓ q » H 0 pU p b C p , Q ℓ q _ .
And since M an 0 is connected, we can insure that all the U 's involved are connected as well. Therefore H 0 pU p b C p , Q ℓ q » Q ℓ , and all the transition maps in the direct limit are identity. It follows that H 2pn´1q c pM an 0 , Q ℓ q is trivial.

3.4.2 Compactly induced representations and type theory 3.4.2.1 Let ReppJq denote the category of smooth Q ℓ -representations of G. Let χ be a continuous character of the center ZpJq » Q p2 and let V P ReppJq. We define the maximal quotient of V on which the center acts like χ as follows. Let us consider the set Ω :" tW | W is a subrepresentation of V and ZpJq acts like χ on V {W u.

The set Ω is stable under arbitrary intersection, so that W ˝:" Ş W PΩ W P Ω. The maximal quotient is defined by

V χ :" V {W ˝.
It satisfies the following universal property.

Proposition. Let χ be a continuous character of ZpJq and let V, V 1 P ReppJq. Assume that ZpJq acts like χ on V 1 . Then any morphism V Ñ V 1 factors through V χ .

Proof. Let f : V Ñ V 1 be a morphism of J-representations. Since V {Kerpf q » Impf q Ă V 1 , the center ZpJq acts like χ on the quotient V {Kerpf q. Therefore Kerpf q P Ω. It follows that Kerpf q contains W ˝and as a consequence, f factors through V χ .

3.4.2.2

As representations of J, the terms E a,b 1 of the spectral sequence 3.4.1.4 consist of representations of the form c ´Ind J J θ ρ, where ρ is the inflation to J θ of a representation of the finite group of Lie type J θ . We note that such a compactly induced representation does not contain any smooth irreducible subrepresentation of J. Indeed, the center ZpJq » Q p2 does not fix any finite dimensional subspace. In order to rectify this, it is customary to fix a continuous character χ of ZpJq which agrees with the central character of ρ on ZpJq X J θ » Z p2 , and to describe the space pc ´Ind J J θ ρq χ instead.

Lemma. We have pc ´Ind J J θ ρq χ » c ´Ind J ZpJqJ θ χ b ρ.

Proof. By Frobenius reciprocity, the identity map on c ´Ind J ZpJqJ θ χ b ρ gives a morphism χ b ρ Ñ `c ´Ind J ZpJqJ θ χ b ρ ˘|ZpJqJ θ of ZpJqJ θ -representations. Restricting further to J θ , we obtain a morphism ρ Ñ `c ´Ind J ZpJqJ θ χ b ρ ˘|J θ . By Frobenius reciprocity, this corresponds to a morphism c ´Ind J J θ ρ Ñ c ´Ind J ZpJqJ θ χ b ρ of J-representations. Because ZpJq acts via the character χ on the target space, this morphism factors through a map pc ´Ind J J θ ρq χ Ñ c ´Ind J ZpJqJ θ χ b ρ. In order to prove that this is an isomorphism, we build its inverse. The quotient morphism c ´Ind J J θ ρ Ñ pc ´Ind J J θ ρq χ corresponds, via Frobenius reciprocity, to a morphism ρ Ñ pc ´Ind J J θ ρq χ |J θ of J θ -representations. Because ZpJq acts via the character χ on the target space, this arrow may be extended to a morphism χbρ Ñ pc ´Ind J J θ ρq χ |ZpJqJ θ of ZpJqJ θrepresentations. By Frobenius reciprocity, this corresponds to a morphism c ´Ind J ZpJqJ θ χbρ Ñ pc ´Ind J J θ ρq χ , and this is our desired inverse.

3.4.2.3

We recall a general theorem from [START_REF] Bushnell | Induced representations of locally profinite groups[END_REF] describing certain compactly induced representations. In this paragraph only, let G be any p-adic group, and let L be an open subgroup of G which contains the center ZpGq and which is compact modulo ZpGq.

Theorem ([Bus90] Theorem 2 (supp)). Let pσ, V q be an irreducible smooth representation of L. There is a canonical decomposition

c ´Ind G L σ » V 0 ' V 8
, where V 0 is the sum of all supercuspidal subrepresentations of c ´Ind G L σ, and where V 8 contains no non-zero admissible subrepresentation. Moreover, V 0 is a finite sum of irreducible supercuspidal subrepresentations of G.

The spaces V 0 or V 8 could be zero. Note also that since G is p-adic, any irreducible representation is admissible. So in particular, V 8 does not contain any irreducible subrepresentation. However, it may have many irreducible quotients and subquotients. Thus, the space V 8 is in general not G-semisimple. Hence, the structure of the compactly induced representation c ´Ind G L σ heavily depends on the supercuspidal supports of its irreducible subquotients. We go back to our previous notations. Let 0 ď θ ď m, let ρ be a smooth irreducible representation of J θ and let χ be a character of ZpJq agreeing with the central character of ρ on ZpJq X J θ . Since the group ZpJqJ θ contains the center and is compact modulo the center, we have a canonical decomposition pc ´Ind J J θ ρq χ » V ρ,χ,0 ' V ρ,χ,8 . In order to describe the spaces V ρ,χ,0 and V ρ,χ,8 , we determine the supercuspidal supports of the irreducible subquotients of c ´Ind J J θ ρ through type theory, with the assumption that ρ is inflated from J θ . For our purpose, it will be enough to analyze only the case θ " m. In this case, dim V 1 m is equal to 0 or 1 so that GUpV 1 m q " t1u or F p2 has no proper parabolic subgroup. In particular, if ρ is a cuspidal representation of GUpV 0 m q, then its inflation to the reductive quotient J m » GpUpV 0 m q ˆUpV 1 m qq is also cuspidal.

3.4.2.4

In the following paragraphs, we recall a few general facts from type theory. For more details, we refer to [START_REF] Bushnell | Smooth representations of reductive p-adic groups: structure theory via types[END_REF] and [START_REF] Morris | Level-0 G-types[END_REF]. Let G be the group of F -rational points of a reductive connected group G over a p-adic field F . A parabolic subgroup P (resp. Levi complement L) of G is defined as the group of F -rational points of an F -rational parabolic subgroup P Ă G (resp. an F -rational Levi complement L Ă G). Every parabolic subgroup P admits a Levi decomposition P " LU where U is the unipotent radical of P . We denote by X F pGq the set of F -rational Q ℓ -characters of G, and by X un pGq the set of unramified characters of G, ie. the continuous characters of G which are trivial on all compact subgroups. We consider pairs pL, τ q where L is a Levi complement of G and τ is a supercuspidal representation of L. Two pairs pL, τ q and pL 1 , τ 1 q are said to be inertially equivalent if for some g P G and χ P X un pGq we have L 1 " L g and τ 1 » τ g b χ where τ g is the representation of L g defined by τ g plq :" τ pg ´1lgq. This is an equivalence relation, and we denote by rL, τ s G or rL, τ s the inertial equivalence class of pL, τ q in G. The set of all inertial equivalence classes is denoted ICpGq. If P is a parabolic subgroup of G, we write ι G P for the normalised parabolic induction functor. Any smooth irreducible representation π of G is isomorphic to a subquotient of some parabolically induced representation ι G P pτ q where P " LU for some Levi complement L and τ is a supercuspidal representation of L. We denote by ℓpπq P ICpGq the inertial equivalence class rL, τ s. This is uniquely determined by π and it is called the inertial support of π.

3.4.2.5 Let s P ICpGq. We denote by Rep s pGq the full subcategory of ReppGq whose objects are the smooth representations of G all of whose irreducible subquotients have inertial support s. This definition corresponds to the one given in [START_REF] Bernstein | Le centre de Bernstein[END_REF] 2.8. If S Ă ICpGq, we write Rep S pGq for the direct product of the categories Rep s pGq where s runs over S. We recall the main results from loc. cit.

Theorem ([BD84] 2.8 and 2.10). The category ReppGq decomposes as the direct product of the subcategories Rep s pGq where s runs over ICpGq. Moreover, if S Ă ICpGq then the category Rep S pGq is stable under direct sums and subquotients.

Type theory was then introduced in [START_REF] Bushnell | Smooth representations of reductive p-adic groups: structure theory via types[END_REF] in order to describe the categories Rep s pGq which are called the Bernstein blocks.

3.4.2.6

Let S be a subset of ICpGq. A S-type in G is a pair pK, ρq where K is an open compact subgroup of G and ρ is a smooth irreducible representation of K, such that for every smooth irreducible representation π of G we have

π |K contains ρ ðñ ℓpπq P S.
When S is a singleton tsu, we call it an s-type instead.

Remark. By Frobenius reciprocity, the condition that π |K contains ρ is equivalent to π being isomorphic to an irreducible quotient of c ´Ind G K ρ. In fact, we can say a little bit more. Let K be an open compact subgroup of G and let ρ be an irreducible smooth representation of K. Let Rep ρ pGq denote the full subcategory of ReppGq whose objects are those representations which are generated by their ρ-isotypic component. If pK, ρq is an S-type, then [START_REF] Bushnell | Smooth representations of reductive p-adic groups: structure theory via types[END_REF] Theorem 4.3 establishes the equality of categories Rep ρ pGq " Rep S pGq. By definition of compact induction, the representation c ´Ind G K ρ is generated by its ρ-isotypic vectors. Therefore any irreducible subquotient of c ´Ind G K ρ has inertial support in S.

3.4.2.7 An important class of types are those of depth zero, and they are the only ones we shall encounter. First, we recall the following result. If K is a parahoric subgroup of G, we denote by K its maximal reductive quotient. It is a finite group of Lie type over the residue field of F .

Proposition ([Mor99] 4.1). Let K be a maximal parahoric subgroup of G and let ρ be an irreducible cuspidal representation of K. We see ρ as a representation of K by inflation. Let π be an irreducible smooth representation of G and assume that π |K contains ρ. Then π is supercuspidal and there exists an irreducible smooth representation ρ of the normalizer N G pKq such that ρ|K contains ρ and π » c ´Ind G N G pKq ρ.

Such representations π are called depth-0 supercupidal representations of G. More generally, a smooth irreducible representation π of G is said to be of depth-0 if it contains a non-zero vector that is fixed by the pro-unipotent radical of some parahoric subgroup of G. A depth-0 type in G is a pair pK, ρq where K is a parahoric subgroup of G and ρ is an irreducible cuspidal representation of K, inflated to K. The name is justified by the following theorem.

Theorem ([Mor99] 4.8). Let pK, ρq be a depth-0 type. Then there exists a (unique) finite set S Ă ICpGq such that pK, ρq is an S-type of G.

In loc. cit. it is also proved that any depth-0 supercuspidal representation of G contains a unique conjugacy class of depth-0 types. Let K be a parahoric subgroup of G. Using the Bruhat-Tits building of G, one may canonically associate a Levi complement L of G such that K L :" L X K is a maximal parahoric subgroup of L, whose maximal reductive quotient K L is naturally identified with K. This is precisely described in [Mor99] 2.1. Moreover, we have L " G if and only if K is a maximal parahoric subgroup of G. Now, let pK, ρq be a depth-0 type of G and denote by S the finite subset of ICpGq such that it is an S-type of G. Since ρ is a cuspidal representation of K » K L , we may inflate it to K L . Then, the pair pK L , ρq is a depth-0 type of L. We say that pK, ρq is a G-cover of pK L , ρq. By the previous theorem, there is a finite set S L Ă ICpLq such that pK L , ρq is an S L -type of L. Then the proof of Theorem 4.8 in [START_REF] Morris | Level-0 G-types[END_REF] shows that we have the relation

S " rM, τ s G ˇˇrM, τ s L P S L ( .
In this set, M is some Levi complement of L, therefore it may also be seen as a Levi complement in G. Thus, an inertial equivalence class rM, τ s L in L gives rise to a class rM, τ s G in G.

Since K L is maximal in L, in virtue of the proposition above any element of S L has the form rL, πs L for some supercuspidal representation π of L. In particular, every smooth irreducible representation of G containing the type pK, ρq has a conjugate of L as cuspidal support. We deduce the following corollary.

Corollary. Let pK, ρq be a depth-0 type in G and assume that K is not a maximal parahoric subgroup. Then no smooth irreducible representation π of G containing the type pK, ρq is supercuspidal.

3.4.2.8 Thus, up to replacing G with a Levi complement, the study of any depth-0 type pK, ρq can be reduced to the case where K is a maximal parahoric subgroup. Let us assume that it is the case, and let S be the associated finite subset of ICpGq. While S is in general not a singleton, it becomes one once we modify the pair pK, ρq a little bit. Let p K be the maximal open compact subgroup of N G pKq. We have K Ă p K but in general this inclusion may be strict. Let ρ be a smooth irreducible representation of N G pKq such that ρ|K contains ρ. Let p ρ be any irreducible component of the restriction ρ| p K . Eventually, let π :" c ´Ind G N G pKq ρ be the associated depth-0 supercuspidal representation of G.

Theorem ([Mor99] Variant 4.7). The pair p p K, p ρq is a rG, πs-type.

The conclusion does not depend on the choice of p ρ as an irreducible component of ρ| p K . Any one of them affords a type for the same singleton s " rG, πs.

3.4.2.9

Let us now consider a parahoric subgroup K along with an irreducible representation ρ of its maximal reductive quotient K " K{K `, where K `is the pro-unipotent radical of K. Assume that ρ is not cuspidal. Thus, there exists a proper parabolic subgroup P Ă K with Levi complement L, and a cuspidal irreducible representation τ of L, such that ρ is an irreducible component of the Harish-Chandra induction ι K P τ . The preimage of P via the quotient map K ։ K is a parahoric subgroup K 1 Ĺ K, whose maximal reductive quotient K 1 :" K 1 {K 1`i s naturally identified with L. We have K `Ă K 1`Ă K 1 and the intermediate quotient K 1`{ K `is identified with the unipotent radical N of P » K 1 {K `. Consider ρ as an irreducible representation of K inflated from K. The invariants ρ K 1`f orm a representation of K 1 which coincides with the inflation of the Harish-Chandra restriction of ρ (as a representation of K) to L. Thus, ρ K 1`c ontains the inflation of τ to a representation of K 1 . In other words, we have a K 1 -equivariant map τ Ñ ρ |K 1 .

By Frobenius reciprocity, it gives a map

c ´Ind K K 1 τ Ñ ρ,
which is surjective by irreducibility of ρ. Applying the functor c ´Ind G K : ReppKq Ñ ReppGq, which is exact, and using transitivity of compact induction, we deduce the existence of a natural surjection c ´Ind G K 1 τ ։ c ´Ind G K ρ. Now, pK 1 , τ q is a depth-0 type in G. Let S Ă ICpGq be the subset such that pK 1 , τ q is an S-type, and let L be the (proper) Levi complement of G associated to K 1 as in the previous paragraph. By 3.4.2.6 Remark, it follows that any irreducible subquotient of c ´Ind G K ρ has inertial support in S. Since all elements of S are of the form rL, πs for some supercuspidal representation π of L, we reach the following conclusion. which case J 0 " J ˝and ZpJqJ 0 is of index 2 in N J pJ 0 q " J. A representative of the non-trivial coset is given by g 0 as defined in 3.1.1.7. If n " 2, define τ m,χ :" c ´Ind J ZpJqJm χ b ρ λ .

Then τ m,χ is an irreducible supercuspidal representation of J, and we have

`c ´Ind J Jm ρ λ ˘χ » c ´Ind J ZpJqJm χ b ρ λ " τ m,χ .
Thus V ρ λ ,χ,8 " 0 and V ρ λ ,χ8 " τ m,χ in this case.

When n " 2, ρ λ " ρ ∆ 0 " 1 is the trivial representation of J 0 " J ˝. Let χ 0 : J Ñ Q ℓ ˆbe the unique non-trivial character of J which is trivial on ZpJqJ 0 . Then `c ´Ind J J 0 1 ˘χ is the sum of an unramified character τ 0,χ of J whose central character is χ, and of the character χ 0 τ 0,χ . Both characters are supercuspidal, and they are the only unramified characters of J with central character χ.

3.4.2.12 According to 3.4.1.6 and 3.4.1.11, the terms E 0,b 1 are a sum of representations of the form c ´Ind J Jm ρ λ , with λ a partition of 2m `1 having 2-core ∆ 0 if b is even, and ∆ 1 if b is odd. Moreover, by 3.4.1.11 we have

E 0,2pn´1´mq 2 » c ´Ind J Jm 1, E 0,2pn´1´mq`1 2 » c ´Ind J Jm ρ p2m,1q .
In particular, summing up the discussion of the previous paragraph, we have reached the following statement.

Proposition. Let χ be an unramified character of ZpJq.

-Assume that n ě 3. The representation pE 0,2pn´1´mq 2 q χ contains no non-zero admissible subrepresentation, and it is not J-semisimple. Moreover, any irreducible subquotient has inertial support rL 0 , τ 0 s. If n ě 5, then the same statement holds for pE 0,2pn´1´mq`1 2 q χ with the inertial support being rL 1 , τ 1 s.

-For n " 1, 2, 3, 4, let b " 0, 2, 3, 5 respectively. Then m " 0 when 1, 2 and m " 1 when n " 3, 4. Let χ be an unramified character of ZpJq. The representation τ m,χ is irreducible supercuspidal, and we have

pE 0,b 2 q χ » $ & % τ m,χ if n " 1, 3, 4, τ m,χ ' χ 0 τ m,χ if n " 2.
In particular, we deduce the following important corollary.

Corollary. Let χ be an unramified character of ZpJq. If n ě 3 then H 2pn´1´mq c pM an , Q ℓ q χ is not J-admissible. If n ě 5 then the same holds for H 2pn´1´mq`1 c pM an , Q ℓ q χ . 3.4.3 The case n " 3, 4 3.4.3.1 Let us focus on the case m " 1, that is n " 3 or 4. Recall that N pΛ 0 q denotes the set of lattices Λ P L 0 with type tpΛq " t max " 3 containing Λ 0 . It has cardinality νp1, 2q " p `1 when n " 3 and νp2, 3q " p 3 `1 when n " 4. In particular, we may locate the non zero terms E a,b 1,alt of the alternating Čech spectral sequence as follows.

E a,b 1,alt " 0 ðñ $ & % pa, bq P tp0, 2q; p0, 3q; p´k, 4q | 0 ď k ď pu if n " 3, pa, bq P tp0, 4q; p0, 5q; p´k, 6q | 0 ď k ď p 3 u if n " 4.

In Figure 3 below, we draw the shape of the first page E 1,alt for n " 3. The case of n " 4 is similar, except that two more 0 rows should be added at the bottom. To alleviate the notations, we write ϕ ´a for the differential ϕ

2pn´1q ´a . . . . E ´3,4 1,alt E ´2,4 1,alt E ´1,4 1,alt c ´Ind J J 1 1 c ´Ind J J 1 ρ ∆ 2 c ´Ind J J 1 1 0 0 ϕ 4 ϕ 3 ϕ 2 ϕ 1
Figure 3: The first page E 1,alt of the alternating Čech spectral sequence when n " 3.

3.4.3.2 Let i P Z such that ni is even. For Λ, Λ 1 P L i , recall that the distance dpΛ, Λ 1 q is the smallest integer d ě 0 such that there exists a sequence Λ " Λ 0 , . . . , Λ d " Λ 1 of lattices of L i with tΛ j , Λ j`1 u being an edge for all 0 ď j ď d ´1. When m " 1, any lattice Λ P L i has type 1 or 3, and two lattices forming an edge can not have the same type. Therefore, the value of tpΛ j q alternates between 1 and 3. In particular, if tpΛq " tpΛ 1 q then dpΛ, Λ 1 q is even. According to [START_REF] Vollaard | The Supersingular Locus of the Shimura Variety for GU(1, s)[END_REF] Proposition 3.7, the simplicial complex L i is in fact a tree. We will use this to prove the following proposition. 2 " Kerpϕ 1 q{Impϕ 2 q vanishes. As we have observed in 3.4.1.12, the term E a,b 1 is the Q ℓ -vector space generated by the set I ´a`1 , and E a,b 1,alt is the subspace consisting of all the vectors v " ř γPI ´a`1 λ γ γ such that for all σ P S ´a`1 we have λ σpγq " sgnpσqλ γ . Here the λ γ 's are scalars which are almost all zero. To prove the proposition, let us look at the differential ϕ 2 . It acts on the basis vectors in the following way.

pΛ, Λ, Λq pΛ, Λ, Λ 1 q pΛ 1 , Λ, Λq , / .

/ -Þ Ñ pΛ, Λq, @Λ,

Λ 1 P L p1q such that U Λ X U Λ 1 " H, pΛ, Λ 1 , Λq Þ Ñ pΛ 1 , Λq `pΛ, Λ 1 q ´pΛ, Λq, @Λ, Λ 1 P L p1q such that U Λ X U Λ 1 " H, pΛ, Λ 1 , Λ 2 q Þ Ñ pΛ, Λ 1 q `pΛ 1 , Λ 2 q ´pΛ, Λ 2 q, @Λ, Λ 1 , Λ 2 P L p1q such that U Λ X U Λ 1 X U Λ 2 " H.
We note that for a collection of lattices Λ 1 , . . . , Λ s P L p1q i , the condition U Λ 1 X . . . X U Λ s " H is equivalent to dpΛ j , Λ j 1 q " 2 for all 1 ď j " j 1 ď s. Towards a contradiction, we assume that Impϕ 2 q Ĺ Kerpϕ 1 q. Let v P Kerpϕ 1 qzImpϕ 2 q. Since v P E ´1,b 1,alt , it decomposes under the form v " r ÿ j"1 λ j pγ j ´τ pγ j qq, where r ě 1, the γ j 's are of the form pΛ, Λ 1 q with Λ " Λ 1 and U Λ X U Λ 1 " H, the scalars λ j 's are non zero and τ P S 2 is the transposition. We may assume that r is minimal among all the vectors in the complement Kerpϕ 1 qzImpϕ 2 q. In particular, there exists a single i P Z such that ni is even, and for all j the lattices in γ j belong to L p1q i . We may further assume i " 0 without loss of generality. We say that an element γ P I 2 occurs in v if γ " γ j or τ pγ j q for some j. Similarly, we say that a lattice Λ P L p1q 0 occurs in v if it is a constituent of some γ j .

Lemma. Let γ " pΛ 1 , Λq P I 2 be an element occuring in v. Then there exists Λ 2 P L p1q 0 such that pΛ 2 , Λq P I 2 occurs in v and dpΛ 1 , Λ 2 q " 4.

Proof. Let us write pΛ j , Λq P I 2 , 1 ď j ď s for the various elements occuring in v whose first component is Λ. Up to reordering the γ j 's and swapping them with τ pγ j q if necessary, we may assume that pΛ j , Λq " γ j for all 1 ď j ď s, and that Λ 1 " Λ 1 . The coordinate of ϕ 1 pvq along the basis vector pΛq is equal to 2 ř s j"1 λ j . Since ϕ 1 pvq " 0, the sum of the λ j 's from 1 to s is zero. In particular, we have s ě 2. For all 2 ď j ď s, we have 2 ď dpΛ 1 , Λ j q ď 4 by the triangular inequality. Towards a contradiction, assume that dpΛ 1 , Λ j q " 2 for all 2 ď j ď s. In particular, δ j :" pΛ j , Λ 1 , Λq P I 3 for all 2 ď j ď s. Consider the vector w :" 1 3 s ÿ j"2 ÿ σPS 6 sgnpσqλ j σpδ j q P E ´2,b 1,alt .

Then we compute ϕ 2 pwq " ´λ1 ppΛ 1 , Λq ´pΛ, Λ 1 qq ´s ÿ j"2 λ j ppΛ j , Λq ´pΛ, Λ j qq `s ÿ j"2 λ j ppΛ j , Λ 1 q ´pΛ 1 , Λ j qq.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

In particular, we get v `ϕ2 pwq " r ÿ j"s`1 λ j pγ j ´τ pγ j qq `s ÿ j"2 λ j ppΛ j , Λ 1 q ´pΛ 1 , Λ j qq P Kerpϕ 1 qzImpϕ 2 q, which contradicts the minimality of r.

3.4.3.3

To conclude the proof of the proposition, let us pick Λ " Λ 0 P L p1q 0 which occurs in v, say in a pair pΛ 1 , Λq P I 2 . Write Λ 1 :" Λ 1 . By induction, we build a sequence pΛ k q kě0 of lattices in L p1q 0 such that for all k, the pair pΛ k`1 , Λ k q occurs in v and we have dpΛ 0 , Λ k q " 2k. It follows that the Λ k 's are pairwise distinct, and it leads to a contradiction since only a finite number of such lattices can occur in v. Let us assume that Λ 0 , . . . , Λ k are already built for some k ě 1. By the Lemma applied to Λ k , there exists Λ k`1 P L p1q 0 such that the pair pΛ k`1 , Λ k q occurs in v and dpΛ k´1 , Λ k`1 q " 4. By the triangular inequality, we have dpΛ 0 , Λ k`1 q ě |dpΛ 0 , Λ k q ´dpΛ k , Λ k`1 q| " 2k ´2 " 2pk ´1q.

Thus dpΛ 0 , Λ k`1 q " 2pk ´1q, 2k or 2pk `1q. We prove that it must be equal to the latter. Assume dpΛ 0 , Λ k`1 q " 2pk ´1q. There exists a path Λ 0 " L 0 , . . . , L 2pk´1q " Λ k`1 . We obtain a cycle

Λ 0 X Λ 1 Λ 1 . . . Λ k´1 Λ k´1 X Λ k Λ 0 Λ k L 1 L 2 . . . L 2pk´1q " Λ k`1 Λ k X Λ k`1
Since L 0 is a tree, this cycle must be trivial, ie. the lower and upper paths, which are of the same length, are the same. In particular, we have Λ k´1 " Λ k`1 , which is absurd since dpΛ k´1 , Λ k`1 q " 4.

Assume dpΛ 0 , Λ k`1 q " 2k. There exists a path Λ 0 " L 0 , . . . , L 2k " Λ k`1 . We obtain a cycle

Λ 0 X Λ 1 Λ 1 . . . Λ k´1 X Λ k Λ k Λ 0 Λ k X Λ k`1 L 1 L 2 . . . L 2k´1 L 2k " Λ k`1
Since L 0 is a tree, this cycle must be trivial, ie. the lower and upper paths, which are of the same length, are the same. In particular, we have Λ k " Λ k`1 , which is absurd since dpΛ k , Λ k`1 q " 2.

Thus, we have dpΛ 0 , Λ k`1 q " 2pk `1q so that Λ k`1 meets all the requirements. It concludes the proof.

3.4.3.4

In particular, we obtain the following statement.

Theorem. Assume that n " 3 or 4. Let b " 3 if n " 3, and let b " 5 if n " 4. We have

H b c pM an , Q ℓ q » c ´Ind J J 1 ρ ∆ 2 ,
with the rational Frobenius τ acting like multiplication by ´pb .

3.5

The cohomology of the basic stratum of the Shimura variety for n " 3, 4

3.5.1 The Hochschild-Serre spectral sequence induced by p-adic uniformization 3.5.1.1 In this section, we still assume that n is any integer ě 1. We recover the notations of Part 3.3 regarding Shimura varieties. As we have seen in 3.3.6, p-adic uniformization is a geometric identity relating the Rapoport-Zink space M with the basic stratum S K p pb 0 q. In [Far04], Fargues constructed a Hochschild-Serre spectral sequence using the uniformization theorem on the generic fibers, which we introduce in the following paragraphs.

Recall the PEL datum introduced in 3.3.1. Let ξ : G Ñ W ξ be a finite-dimensional irreducible algebraic Q ℓ -representation of G. Such representations have been classified in [HT01] III.2. We look at V Q ℓ :" V b Q ℓ as a representation of G, whose dual is denoted by V 0 . Using the alternating form x¨, ¨y, we have an isomorphism

V 0 » V Q ℓ b c ´1,
where c is the multiplier character of G.

Proposition ([HT01] III.2).

There exists unique integers tpξq, mpξq ě 0 and an idempotent ǫpξq P EndpV bmpξq 0 q such that W ξ » c tpξq b ǫpξqpV bmpξq 0 q.

The weight wpξq is defined by wpξq :" mpξq ´2tpξq.

To any ξ as above, we can associate a local system L ξ which is defined on the tower pS K p q K p of Shimura varieties. We still write L ξ for its restriction to the generic fiber Sh K 0 K p b E Z p 2 , and we denote by L ξ its restriction to the special fiber S K p . Let A K p be the universal abelian scheme over S K p . We write π m K p : A m K p Ñ S K p for the structure morphism of the m-fold product of A K p with itself over S K p . If m " 0 it is just the identity on S K p . According to [HT01] III.2, we have an isomorphism

L ξ » ǫpξqǫ mpξq ´Rmpξq pπ mpξq K p q ˚Qℓ ptpξqq ¯,
where ǫ mpξq is some idempotent. In particular, if ξ is the trivial representation of G then L ξ " Q ℓ .

3.5.1.2

We fix an irreducible algebraic representation ξ : G Ñ W ξ as above. We associate the space A ξ of automorphic forms of I of type ξ at infinity. Explicitly, it is given by A ξ " tf : IpA f q Ñ W ξ | f is IpA f q-smooth by right translations and @γ P IpQq, f pγ ¨q " ξpγqf p¨qu .

We denote by L an ξ the analytification of L ξ to Sh an K 0 K p , as well as for its restriction to any open subspace.

Notation. We write H ' pp p S K p q an |b 0 , L an ξ q for the cohomology of p p S K p q an |b 0 p b C p with coefficients in L an ξ .

Theorem ([Far04] 4.5.12). There is a W -equivariant spectral sequence F a,b 2 pK p q : Ext a J `H2pn´1q´b c pM an , Q ℓ qp1 ´nq, A K p ξ ˘ùñ H a`b pp p S K p q an |b 0 , L an ξ q. These spectral sequences are compatible as the open compact subgroup K p varies in GpA p f q.

The W -action on F a,b 2 pK p q is inherited from the cohomology group H 2pn´1q´b c pM an , Q ℓ qp1 ´nq. By the compatibility with K p , we may take the limit lim Ý ÑK p for all terms and obtain a GpA p f qˆWequivariant spectral sequence. Since m is the semisimple rank of J, the terms F a,b 2 pK p q are zero for a ą m according to [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] Lemme 4.4.12. Therefore, the non-zero terms F a,b 2 are located in the finite strip delimited by 0 ď a ď m and 0 ď b ď 2pn ´1q.

Let us look at the abutment of the sequence. Since the formal completion p S K p of S K p along its special fiber is a smooth formal scheme, Berkovich's comparison theorem ([Ber96] Corollary 3.7) gives an isomorphism H a`b c pS K p pb 0 q b F, L ξ q " H a`b pS K p pb 0 q b F, L ξ q " Ý Ñ H a`b pp p S K p q an |b 0 , L an ξ q. The first equality follows from S K p pb 0 q being a proper variety. Since this variety has dimension m, the cohomology H ' pp p S K p q an |b 0 , L an ξ q is concentrated in degrees 0 to 2m.

3.5.1.3 Let ApIq denote the set of all automorphic representations of I counted with multiplicities. We write q ξ for the dual of ξ. We also define A ξ pIq :" tΠ P ApIq | Π 8 " q ξu.

According to [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] 4.6, we have an identification

A Kp ξ » à ΠPA ξ pIq Π p b pΠ p q Kp .
It yields, for every a and b, an isomorphism

F a,b 2 pK p q » à ΠPA ξ pIq Ext a J `H2pn´1q´b c pM an , Q ℓ qp1 ´nq, Π p ˘b pΠ p q Kp .
Taking the limit over K p , we deduce that

F a,b 2 :" lim Ý Ñ K p F a,b 2 pK p q » à ΠPA ξ pIq Ext a J `H2pn´1q´b c pM an , Q ℓ qp1 ´nq, Π p ˘b Π p .
The spectral sequence defined by the terms F a,b 2 computes H a`b p p S an |b 0 , L an ξ q :" lim Ý ÑK p H a`b pp p S K p q an |b 0 , L an ξ q. It is isomorphic to H a`b c pSpb 0 q b F, L ξ q :" lim Ý ÑK p H a`b c pS K p pb 0 q b F, L ξ q.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q 3.5.1.4 Recall from 3.4.1.7 that we have a decomposition

H b c pM an , Q ℓ q » à bďb 1 ď2pn´1q E b´b 1 ,b 1 2
, and E b´b 1 ,b 1 2 corresponds to the eigenspace of τ associated to the eigenvalue p´pq b . Accordingly, we have a decomposition

F a,b 2 » à 2pn´1q´b ď b 1 ď 2pn´1q à ΠPA ξ pIq Ext a J ´E2pn´1q´b´b 1 ,b 1 2 p1 ´nq, Π p ¯b Π p .
For Π P A ξ pIq, we denote by ω Π the central character. We define

δ Πp :" ω Πp pp ´1 ¨idqp ´wpξq P Q ℓ ˆ.
Let ι be any isomorphism Q ℓ » C, and write | ¨|ι :" |ιp¨q|. Since I is a group of unitary similitudes of an E{Q-hermitian space, its center is E ˆ¨id. The element p ´1 ¨id P ZpJq can be seen as the image of p ´1 ¨id P ZpIpQqq. We have ω Π pp ´1 ¨idq " 1. Moreover, for any finite place q " p, the element p ´1 ¨id lies inside the maximal compact subgroup of ZpIpQ q qq, so |ω Πq pp ´1idq| ι " 1. Besides Π 8 " q ξ, so we have

|ω Πp pp ´1 ¨idq| ι " |ω q ξ pp ´1 ¨idq| ´1 ι " |ω ξ pp ´1 ¨idq| ι " |p wpξq | ι " p wpξq .
The last equality comes from the isomorphism W ξ » c tpξq b ǫpξqpV bmpξq 0 q, see 3.5.1.1. In particular |δ Πp | ι " 1 for any isomorphism ι.

Proposition. The W -action on Ext a J pE 2pn´1q´b´b 1 ,b 1 2 p1 ´nq, Π p q is trivial on the inertia I, and the Frobenius element Frob acts like multiplication by p´1q ´b1 δ Πp p ´b1 `2pn´1q`wpξq .

Proof. Let us write X :" E 2pn´1q´b´b 1 ,b 1 2 p1 ´nq. By convention, the action of Frob on a space Ext a J pX, Π p q is induced by functoriality of Ext applied to Frob ´1 : X Ñ X. Let us consider a projective resolution of X in the category of smooth representations of J . . .

P 2 P 1 P 0 X 0. u 3 u 2 u 1 u 0
Since Frob ´1 commutes with the action of J, we can choose a lift F " pF i q iě0 of Frob ´1 to a morphism of chain complexes.

. . .

P 2 P 1 P 0 X 0 . . . P 2 P 1 P 0 X 0 u 3 u 2 F 2 u 1 F 1 u 0 F 0 Frob ´1 u 3 u 2 u 1 u 0
After applying Hom J p¨, Π p q and forgetting about the first term, we obtain a morphism F ˚of chain complexes. 0 Hom J pP 0 , Π p q Hom J pP 1 , Π p q Hom J pP 2 , Π p q . . . 0 Hom J pP 0 , Π p q Hom J pP 1 , Π p q Hom J pP 2 , Π p q . . .

F 0 F 1 F
Here F i f pvq :" f pF i pvqq. It induces morphisms on the cohomology

F i : Ext i J pX, Π p q Ñ Ext i J pX, Π p q,
which do not depend on the choice of the lift F. Recall that Frob is the composition of τ and p ¨id P J. Since τ is multiplication by the scalar p´1q b 1 p b 1 ´2pn´1q on X, we may choose the lift F i :" p´1q ´b1 p ´b1 `2pn´1q pp ´1 ¨idq for all i.

Consider an element of Ext i J pX, Π p q represented by a morphism f : P i Ñ Π p . For any v P P i we have

F i f pvq " f pF i pvqq " p´1q ´b1 p ´b1 `2pn´1q f ppp ´1 ¨idq ¨vq " p´1q ´b1 p ´b1 `2pn´1q ω Πp pp ´1 ¨idqf pvq.
It follows that Frob acts on Ext i J pX, Π p q via multiplication by the scalar p´1q ´b1 δ Πp p ´b1 `2pn´1q`wpξq .

3.5.1.5 In general, the Hochschild-Serre spectral sequence has many differentials between non-zero terms. However, focusing on the diagonal defined by a `b " 0, it is possible to compute H 0 c pSpb 0 q b F, L ξ q. Recall that X un pJq denotes the set of unramified characters of J. If x P Q ℓ ˆis any non-zero scalar, we denote by Q ℓ rxs the 1-dimensional representation of W where the inertia I acts trivially and the geometric Frobenius Frob acts like x ¨id.

Proposition. We have an isomorphism of GpA p f q ˆW -representations

H 0 c pSpb 0 q b F, L ξ q » à ΠPA ξ pIq ΠpPX un pJq Π p b Q ℓ rδ Πp p wpξq s.
Proof. The only non-zero term F a,b 2 on the diagonal defined by a `b " 0 is F 0,0 2 . Since there is no non-zero arrow pointing at nor coming from this term, it is untouched in all the successive pages of the sequence. Therefore we have an isomorphism

F 0,0 2 » H 0 c pSpb 0 q b F, L ξ q.
Using 3.4.1.12, we also have isomorphisms

F 0,0 2 » à ΠPA ξ pIq Hom J `H2pn´1q c pM an , Q ℓ qp1 ´nq, Π p ˘b Π p » à ΠPA ξ pIq Hom J `pc ´Ind J J ˝1qp1 ´nq, Π p ˘b Π p » à ΠPA ξ pIq Hom J ˝`1p1 ´nq, Π p|J ˝˘b Π p .
Thus, only the automorphic representations Π P A ξ pIq with Π J p " 0 contribute to the sum. Consider such a Π. The irreducible representation Π p is generated by a J ˝-invariant vector. Since J ˝is normal in J, the whole representation Π p is trivial on J ˝. Thus, it is an irreducible representation of J{J ˝» Z. Therefore, it is one-dimensional. Since J ˝is generated by all compact subgroups of J, it follows that Π J p " 0 ðñ Π p P X un pJq. When it is satisfied, the W -representation V 0 Π :" Hom J ˝p1p1 ´nq, Π p q has dimension one and the Frobenius action was described in 3.5.1.4.

3.5.2

The case n " 3, 4 3.5.2.1 In this section, we assume that m " 1, ie. n " 3 or 4. We recover the notations of 3.4.3.1. We use our knowledge so far on the cohomology of the Rapoport-Zink space to entirely compute the cohomology of the basic locus of the Shimura variety via p-adic uniformization. Let ξ be an irreducible finite dimensional algebraic representation of G as in 3.5.1.1. When n " 3 or 4, the semisimple rank of J is m " 1, therefore the terms F a,b 2 are zero for a ą 1. In particular, the spectral sequence degenerates on the second page. Since it computes the cohomology of the basic locus Spb 0 q which is 1-dimensional, we also have F 0,b 2 " 0 for b ě 3, and F 1,b 2 " 0 for b ě 2. In Figure 4, we draw the second page F 2 and we write between brackets the complex modulus of the possible eigenvalues of Frob on each term under any isomorphism ι : Q ℓ » C, as computed in 3.5.1.4.

Remark. The fact that no eigenvalue of complex modulus p wpξq appears in F 0,1 2 nor in F 1,1 2 follows from 3.4.3.2 Proposition, where we proved that E ´1,b 2 " 0 for b " 4 (resp. 6) when n " 3 (resp. 4). F 0,2 2 rp wpξq`2 , p wpξq s 0 F 0,1 2 rp wpξq`1 s F 1,1 2 rp wpξq`1 s F 0,0 2 rp wpξq s F 1,0 2 rp wpξq s Figure 4: The second page F 2 with the complex modulus of possible eigenvalues of Frob on each term.

Proposition. We have F 1,1 2 " 0 and the eigenspaces of Frob on F 0,2 2 attached to any eigenvalue of complex modulus p wpξq are zero.

Proof. By the machinery of spectral sequences, there is a GpA p f q ˆW -subspace of H 2 c pSpb 0 q b F, L ξ q isomorphic to F 1,1 2 , and the quotient by this subspace is isomorphic to F 0,2 2 . We prove that all eigenvalues of Frob on H 2 c pSpb 0 qb F, L ξ q have complex modulus p wpξq`2 . The proposition then readily follows. We need the Ekedahl-Oort stratification on the basic stratum of the Shimura variety. Let K p Ă GpA p f q be small enough. In [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 3.3 and 6.3, the authors define the Ekedahl-Oort stratification on M red and on S K p pb 0 q respectively, and they are compatible via the p-adic uniformization isomorphism. For n " 3 or 4, the stratification on the basic stratum take the following form S K p pb 0 q " S K p r1s \ S K p r3s.

The stratum S K p r1s is closed and 0-dimensional, whereas the other stratum S K p r3s is open, dense and 1-dimensional. In particular, we have a Frobenius equivariant isomorphism between the cohomology groups of highest degree

H 2 c pS K p pb 0 q b F, L ξ q » H 2 c pS K p r3s b F, L ξ q.
Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

According the [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF] 5.3, the closed Bruhat-Tits strata M Λ and S K p ,Λ,k also admit an Ekedahl-Oort stratification of a similar form, and we have a decomposition

S K p r3s " ğ Λ,k S K p ,Λ,k r3s
into a finite disjoint union of open and closed subvarieties. As a consequence, we have the following Frobenius equivariant isomorphisms

H 2 c pS K p r3s b F, L ξ q » à Λ,k H 2 c pS K p ,Λ,k r3s b F, L ξ q » à Λ,k H 2 c pS K p ,Λ,k b F, L ξ q
where the last isomorphism between cohomology groups of highest degree follows from the stratification on the closed Bruhat-Tits strata S K p ,Λ,k . Now, recall from 3.5.1.1 that the local system L ξ is given by L ξ » ǫpξqǫ mpξq ´Rmpξq pπ mpξq K p q ˚Qℓ ptpξqq ¯.

It implies that L ξ is pure of weight wpξq. Since the variety S K p ,Λ,k is smooth and projective, it follows that all eigenvalues of Frob on the cohomology group H 2 c pS K p ,Λ,k b F, L ξ q must have complex modulus p wpξq`2 under any isomorphism ι : Q ℓ » C. The result follows by taking the limit over K p .

In this paragraph, let us compute the term

F 1,0 2 » à ΠPA ξ pIq Ext 1 J `H2pn´1q c pM an , Q ℓ qp1 ´nq, Π p ˘b Π p » à ΠPA ξ pIq Ext 1 J `c ´Ind J J ˝1p1 ´nq, Π p ˘b Π p .
Let St J denote the Steinberg representation of J, and recall that X un pJq denotes the set of unramified characters of J.

Proposition. Let π be an irreducible smooth representation of J. Then

Ext 1 J pc ´Ind J J ˝1, πq " $ & % Q ℓ if Dχ P X un pJq, π » χ ¨St J , 0 otherwise.
In order to prove this proposition, we need a few general facts about restriction of smooth representations to normal subgroups. Let G be a locally profinite group and let H be a closed normal subgroup. If pσ, W q is a representation of H, for g P G we define the representation pσ g , W q by σ g : h Þ Ñ σpg ´1hgq. The representation σ is irreducible if and only if σ g is for any (or for all) g P G.

Lemma. Assume that ZpGqH has finite index in G.

(1) Let π be a smooth irreducible admissible representation of G. There exists a smooth irreducible representation σ of H, an integer r ě 1 and g 1 , . . . , g r P G such that π |H » σ g 1 ' . . . ' σ gr .

Moreover r ď rZpGqH : Gs, and for any g P G there exists some 1 ď i ď r such that σ g » σ g i .

(2) Assume furthermore that G{H is abelian. Let π 1 and π 2 be two smooth admissible irreducible representations of G. The three following statements are equivalent.

pπ 1 q |H » pπ 2 q |H . -There exists a smooth character χ of G which is trivial on H such that π 2 » χ ¨π1 .

-Hom H pπ 1 , π 2 q " 0. (3) Assume that G{H is abelian and that rZpGqH : Gs " 2. Let g 0 P GzZpGqH and let π be a smooth admissible irreducible representation of G. If there exists an irreducible representation σ of H such that π |H » σ ' σ g 0 , then σ fi σ g 0 .

Proof. For (1) and ( 2 Let W denote the underlying vector space of σ. By hypothesis, there exists a linear automorphism f : W Ñ W such that for every h P H and w P W , f pσpg ´1 0 hg 0 q ¨wq " σphq ˝f pwq.

Let us write g 2 0 " z 0 h 0 for some z 0 P ZpGq and h 0 P H. We define ϕ :" f 2 ˝σph 0 q ´1. Then for all h P H and w P W , we have ϕpσphq ¨wq " f 2 pσph ´1 0 hq ¨wq " f 2 pσph ´1 0 hh 0 qσph ´1 0 q ¨wq " f 2 pσpg ´2 0 hg 2 0 qσph ´1 0 q ¨wq " σphq ˝f 2 pσph 0 q ´1 ¨wq " σphq ˝ϕpwq.

Thus ϕ : σ " Ý Ñ σ. By Schur's lemma we have ϕ " λ ¨id for some λ P Q ℓ . Up to replacing f by pχpz 0 qλ ´1q 1{2 f , we may assume that ϕ " χpz 0 q ¨id, ie. f 2 " χpz 0 qσph 0 q. We build a G-representation Π on W which extends σ. Let g P G and define Πpgq "

$ & % χpzqσphq if g " zh P ZpGqH,
χpzqf ˝σphq if g " g 0 zh P g 0 ZpGqH.

Then one may check that Π is a well defined group morphism G Ñ GLpW q. The fact that it is smooth irreducible and admissible follows from Π |H » σ by construction, and it concludes the proof.

Remark. Under the hypotheses of (3), as long as σ is a smooth irreducible admissible representation of H such that σ g 0 » σ and whose central character χ |ZpGqXH can be extended to a character of ZpGq, then one may build Π as in the proof of the lemma.

We may now move on to the proof of the proposition.

Proof. By Frobenius reciprocity we have

Ext 1 J pc ´Ind J J ˝1, πq » Ext 1 J ˝p1, π |J ˝q.
By functoriality of Ext, we have Ext 1 J ˝p1, π |J ˝q " 0 if the central character of π is not unramified. Thus, let us now assume that it is unramified. According to 3.1.3.4, we have J{J ˝» Z, and ZpJqJ ˝" J when n is odd, and is of index 2 in J when n is even. Thus, π |J ˝is irreducible when n is odd, and can either be irreducible, either decompose as σ ' σ g 0 for some irreducible representation σ of J ˝such that σ g 0 fi σ when n is even. Here, g 0 may be defined as in 3.1.1.7. Thus, we are reduced to computing Ext 1 J ˝p1, σq for any irreducible representation σ of J with trivial central character. Let J 1 " UpVq denote the unitary group of V (recall that J " GUpVq is the group of unitary similitudes). Then J 1 is a normal subgroup both of J ˝and of J. Moreover, J ˝{J 1 is isomorphic to the image of the multiplier c |J ˝: J ˝Ñ Z p , in particular it is compact. Thus, we have

Ext 1 J ˝p1, σq » Ext 1 J 1 p1, σ |J 1 q J ˝{J 1 .
Since σ has trivial central character, the J ˝-action on Ext 1 J 1 p1, σ |J 1 q is actually trivial on ZpJ ˝qJ 1 . But this group is equal to the whole of J ˝. Indeed, let g P J ˝. Since Q p 2 {Q p is unramified, there exists some λ P Z p2 such that Normpλq " cpgq. Thus cpλ ´1gq " 1 so that g is the product of λ ¨id P ZpJ ˝q and of an element of J 1 . Hence, J ˝acts trivially on Ext 1 J 1 p1, σ |J 1 q. Since J 1 is an algebraic group, we may use Theorem 2 of [START_REF] Nori | On a duality theorem of Schneider-Stuhler[END_REF], a generalization of a duality theorem of Schneider and Stühler, to finish the computation. Namely, we have

Ext 1 J 1 p1, σ |J 1 q » Hom J 1 pσ |J 1 , Dp1qq _ ,
where D denotes the Aubert-Zelevinsky involution in J 1 . We note that Dp1q " St J 1 is the Steinberg representation of J 1 . Let us justify that the restriction of St J to J 1 is equal to St J 1 . The Steinberg representation St J (resp. St J 1 ) can be characterized as the unique irreducible representation ρ of J (resp. of J 1 ) such that Ext 2 J p1, ρq " 0 (resp. Ext 1 J 1 p1, ρq " 0). The gap between the degrees of the Ext groups for J and for J 1 is explained by the non-compactness of the center of J. Since St J has trivial central character, by [NP20] Proposition 3.4 we have Ext

2 J p1, St J q » Ext 1 J,1 p1, St J q ' Ext 2 J,1 p1, St J q,
where the Ext groups on the right-hand side are taken in the category of smooth representations of J on which the center acts trivially. Equivalently, this is the category of smooth representations of J{ZpJq. Consider the normal subgroup ZpJqJ 1 {ZpJq » J 1 {ZpJq X J 1 " J 1 {ZpJ 1 q, with Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

Theorem. There are GpA p f q ˆW -equivariant isomorphisms

H 0 c pSpb 0 q b F, L ξ q » à ΠPA ξ pIq ΠpPX un pJq Π p b Q ℓ rδ Πp p wpξq s, H 1 c pSpb 0 q b F, L ξ q » à ΠPA ξ pIq DχPX un pJq, Πp"χ¨St J Π p b Q ℓ rδ Πp p wpξq s ' à ΠPA ξ pIq DχPX un pJq, Πp"χ¨τ 1 Π p b Q ℓ r´δ Πp p wpξq`1 s, H 2 c pSpb 0 q b F, L ξ q » à ΠPA ξ pIq Π J 1 p "0 Π p b Q ℓ rδ Πp p wpξq`2 s.
Proof. The statement regarding H 0 c pSpb 0 q b F, L ξ q was already proved in 3.5.1.5. Let us prove the statement regarding H 2 c pSpb 0 q b F, L ξ q first. By 3.5.2.1, we have

H 2 c pSpb 0 q b F, L ξ q » F 0,2 2 » à ΠPA ξ pIq Hom J ´E0,b 2 p1 ´nq, Π p ¯b Π p ,
where b " 2 if n " 3 and b " 4 if n " 4. The term E 0,b 2 is isomorphic to c ´Ind J J 1 1. Therefore, by Frobenius reciprocity we have

Hom J ´E0,b 2 p1 ´nq, Π p ¯» Hom J 1 p1p1 ´nq, Π p q .
Hence, only the automorphic representations Π P A ξ pIq with Π J 1 p " 0 contribute to F 0,2 2 . Such a representation Π p is said to be J 1 -spherical. Since J 1 is a special maximal compact subgroup of J, according to [minguez] 2.1, we have dimpπ J 1 q " 1 for every smooth irreducible J 1 -spherical representation π of J. The result follows using 3.5.1.4 to describe the eigenvalues of Frob.

We now prove the statement regarding H 1 c pSpb 0 q b F, L ξ q. By the Hochschild-Serre spectral sequence, there exists a GpA p f q ˆW -subspace V 1 of this cohomology group such that

V 1 » F 1,0 2 and H 1 c pSpb 0 q b F, L ξ q{V 1 » F 0,1 2 .
We have

F 1,0 2 » à ΠPA ξ pIq Ext 1 J `H2pn´1q c pM an , Q ℓ qp1 ´nq, Π p ˘b Π p » à ΠPA ξ pIq Ext 1 J `c ´Ind J J ˝1p1 ´nq, Π p ˘b Π p » à ΠPA ξ pIq DχPX un pJq, Πp"χ¨St J Π p b Q ℓ rδ Πp p wpξq s,
according to 3.5.2.2, and with the eigenvalues of Frob being given by 3.5.1.4. On the other hand, we have

F 0,1 2 » à ΠPA ξ pIq Hom J ´E0,2pn´1q´1 2 p1 ´nq, Π p ¯b Π p .
By 3.5.1.4, Frob acts on a summand of F 0,1 2 by the scalar ´δΠp p wpξq`1 . Since Frob |V 1 has no eigenvalue of complex modulus p wpξq`1 , the quotient actually splits so that F 0,1 2 is naturally a subspace of H 1 c pSpb 0 q b F, L ξ q. It remains to compute it. We have E 0,2pn´1q´1 2 » c ´Ind J J 1 ρ ∆ 2 , with τ acting like multiplication by ´p3 when n " 3 and by ´p5 when n " 4, and ∆ 2 " p2, 1q is the partition of 2m `1 " 3 defined in 3.2.7. Hence, we have an isomorphism

F 0,1 2 » à ΠPA ξ pIq Hom J `c ´Ind J J 1 ρ ∆ 2 p1 ´nq, Π p ˘b Π p » à ΠPA ξ pIq Hom J 1 `ρ∆ 2 p1 ´nq, Π p|J 1 ˘b Π p .
It follows that only the automorphic representations Π P A ξ pIq whose p-component Π p contains the supercuspidal representation ρ ∆ 2 when restricted to J 1 , contribute to the sum. According to 3.4.2.7, such Π p are precisely those of the form χ ¨τ1 for some χ P X un pJq. By the Mackey formula we have

Hom J `c ´Ind J J 1 ρ ∆ 2 , χ ¨τ1 ˘» Hom J 1 `ρ∆ 2 , τ 1|J 1 » Hom J 1 `ρ∆ 2 , pc ´Ind J N J pJ 1 q Ą ρ ∆ 2 q |J 1 » à hPJ 1 zJ{N J pJ 1 q Hom J 1 X h N J pJ 1 q pρ ∆ 2 , h Ą ρ ∆ 2 q,
where in the last formula we omitted to write the restrictions to J 1 X h N J pJ 1 q. We used the fact that χ |J 1 is trivial. Since Ą ρ ∆ 2 is just the inflation of ρ ∆ 2 from J 1 to N J pJ 1 q " ZpJqJ 1 obtained by letting ZpJq act trivially, we have a bijection

Hom J 1 X h N J pJ 1 q pρ ∆ 2 , h Ą ρ ∆ 2 q » Hom N J pJ 1 qX h N J pJ 1 q pĄ ρ ∆ 2 , h Ą ρ ∆ 2 q.
Now, N J pJ 1 q contains the center, is compact modulo the center, and τ 1 " c ´Ind J N J pJ 1 q Ą ρ ∆ 2 is supercuspidal. It follows that an element h P J intertwines Ą ρ ∆ 2 if and only if h P N J pJ 1 q (see for instance [bushnellbook] 11.4 Theorem along with Remarks 1 and 2). Therefore, only the trivial double coset contributes to the sum and we have

Hom J `c ´Ind J J 1 ρ ∆ 2 , χ ¨τ1 ˘» Hom J 1 pρ ∆ 2 , ρ ∆ 2 q » Q ℓ .
To sum up, we have

F 0,1 2 » à ΠPA ξ pIq DχPX un pJq, Πp"χ¨τ 1 Π p b Q ℓ r´δ Πp p wpξq`1 s.
It concludes the proof.

3.5.3 On the cohomology of the ordinary locus when n " 3 3.5.3.1 In this section, we assume that the Shimura variety is of Kottwitz-Harris-Taylor type.

According to [HT01] I.7, it amounts to assuming that the algebra B from 3.3.1 is a division algebra satisfying a few additional conditions. In particular, B v is either split either a division algebra for every place v of Q, and there must be at least one prime number p 1 (different from p) which splits in E and such that B splits over p 1 . In this situation, the Shimura variety is compact. According to 3.3.5, when n " 3 there is a single Newton stratum other than the basic one. It is the µ-ordinary locus S K p pb 1 q, and it is an open dense subscheme of the special fiber of the Shimura variety. Moreover, since the Shimura variety is compact, the ordinary locus is also an affine scheme according to [goldringnicole] and [koskivirtawedhorn]. By using the spectral sequence associated to the stratification S K p " S K p pb 0 q \ S K p pb 1 q, we may deduce information on the cohomology of the ordinary locus. The spectral sequence is given by

G a,b 1 : H b c pS K p pb a q b F, Q ℓ q ùñ H a`b c pS K p b F, Q ℓ q.
In figure 5, we draw the first page of this sequence.

H 4 c pS K p pb 1 q b F, Q ℓ q H 2 c pS K p pb 0 q b F, Q ℓ q H 3 c pS K p pb 1 q b F, Q ℓ q H 1 c pS K p pb 0 q b F, Q ℓ q H 2 c pS K p pb 1 q b F, Q ℓ q H 0 c pS K p pb 0 q b F, Q ℓ q φ ψ
Figure 5: The first page G 1 .

3.5.3.2 Let v be a place of E above p 1 . The cohomology of the Shimura variety Sh C 0 K p b E E v has been entirely computed in [START_REF] Boyer | Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires[END_REF]. Note that as GpA p f q-representations, the cohomology of

Sh C 0 K p b E E v is isomorphic to the cohomology of Sh C 0 K p b E Q p 2 ,
which in turn is isomorphic to the cohomology of the special fiber S K p using nearby cycles. In particular, we understand perfectly the abutment of the spectral sequence G a,b 1 . Since S K p is smooth and projective, its cohomology admits a symmetry with respect to the middle degree 2. Moreover, by the results of loc. cit. the groups of degree 1 and 3 are zero. It follows that φ is surjective and ψ is injective. Combining with our computations, we deduce the following proposition.

Proposition. There is a GpA p f q ˆW -equivariant isomorphism

H 4 c pSpb 1 q b F, L ξ q » à ΠPA ξ pIq ΠpPX un pJq Π p b Q ℓ rδ Πp p wpξq`4 s.
subgroups of G 1 and those of f pGq. Let P be a parabolic subgroup of G and let P 1 " f pPqZpG 1 q 0 be the corresponding parabolic of G 1 . Then the map gP Þ Ñ f pgPq induces an isomorphism f : X P " Ý Ñ X P 1 which is compatible with the actions of G and G 1 via f . Therefore G and G 1 generate the same Deligne-Lusztig varieties.

4.1.3 Let θ ě 0 and let V be a 2θ-dimensional F q -vector space equipped with a nondegenerate symplectic form p¨, ¨q : V ˆV Ñ F q . Fix a basis pe 1 , . . . , e 2θ q in which p¨, ¨q is described by the matrix ˜0 A θ ´Aθ 0 ¸, where A θ denotes the matrix having 1 on the anti-diagonal and 0 everywhere else. If k is a perfect field extension of F q , let V k :" V b Fq k denote the scalar extension to k equipped with its induced k-symplectic form p¨, ¨q. Let τ : V k " Ý Ñ V k denote the map id b σ. If U Ă V k , let U K denote its orthogonal. We consider the finite symplectic group SppV, p¨, ¨qq » Spp2θ, F q q. It can be identified with G " G F where G is the symplectic group SppV F , p¨, ¨qq » Spp2θ, Fq and F is the Frobenius raising the entries of a matrix to their q-th power. Let T Ă G be the maximal torus of diagonal symplectic matrices and let B Ă G be the Borel subgroup of upper-triangular symplectic matrices. The Weyl system of pT, Bq is identified with pW θ , Sq where W θ is the finite Coxeter group of type B θ and S " ts 1 , . . . , s θ u is the set of simple reflexions. They satisfy the following relations s θ s θ´1 s θ s θ´1 " s θ´1 s θ s θ´1 s θ , s i s i´1 s i " s i´1 s i s i´1 , @ 2 ď i ď θ ´1, s i s j " s j s i , @ |i ´j| ě 2.

Concretely, the simple reflexion s i acts on V by exchanging e i and e i`1 as well as e 2θ´i and e 2θ´i`1 for 1 ď i ď θ ´1, whereas s θ exchanges e θ and e θ`1 . The Frobenius F acts trivially on W θ .

4.1.4

We define the following subset of S I :" ts 1 , . . . , s θ´1 u " Szts θ u.

We consider the generalized Deligne-Lusztig variety X I ps θ q. Since s θ s θ´1 s θ R I, it is not a classical Deligne-Lusztig variety. Let S θ :" X I ps θ q be its closure in G{P I . This normal projective variety occurs as a closed Bruhat-Tits stratum in the special fiber of the ramified unitary PEL Rapoport-Zink space of signature p1, n ´1q, as established in [START_REF] Rapoport | The Supersingular Locus of the Shimura Variety for GU(1,n-1) over a Ramified Prime[END_REF]. In loc. cit. the authors describe the geometry of S θ . We summarize their analysis.

Proposition ([RTW14] 5.3, 5.4). Let k be a perfect field extension of F q . The k-rational points of S θ are given by S θ pkq » tU Ă V k | U K " U and U ď1 Ă U `τ pU qu, where ď1

Ă denotes an inclusion of subspaces with index at most 1. There is a decomposition S θ " X I pidq \ X I ps θ q, where X I pidq is closed and of dimension 0, and X I ps θ q is open, dense of dimension θ. They correspond respectively to points U having U " τ pU q and U Ĺ U `τ pU q. If θ ě 2 then S θ is singular at the points of X I pidq. When θ " 1, we have S 1 » P 1 .

4.1.5 For 0 ď θ 1 ď θ, define I θ 1 :" ts 1 , . . . , s θ´θ 1 ´1u, and w θ 1 :" s θ`1´θ 1 . . . s θ . In particular I 0 " I, I θ´1 " I θ " H, w 0 " id and w 1 " s θ .

Proposition ([RTW14] 5.5). There is a stratification into locally closed subvarieties

S θ " θ ğ θ 1 "0 X I θ 1 pw θ 1 q.
The stratum X I θ 1 pw θ 1 q corresponds to points U such that dimpU `τ pU q`. . .`τ θ 1 `1pU qq " θ `θ1 . The closure in S θ of a stratum X I θ 1 pw θ 1 q is the union of all the strata X It pw t q for t ď θ 1 . The stratum X I θ 1 pw θ 1 q is of dimension θ 1 , and X I θ pw θ q is open, dense and irreducible. In particular S θ is irreducible.

Remark. This stratification plays the role of the Ekedahl-Oort stratification M Λ " Ů t M Λ ptq of the closed Bruhat-Tits strata in the unramified case, see [START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF].

4.1.6 It turns out that the strata X I θ 1 pw θ 1 q are related to Coxeter varieties for symplectic groups of smaller sizes. For 0 ď θ 1 ď θ, define K θ 1 :" ts 1 , . . . s θ´θ 1 ´1, s θ´θ 1 `1, . . . , s θ u " Szts θ´θ 1 u.

Note that K 0 " I 0 " I and K θ " S. We have I θ 1 Ă K θ 1 with equality if and only if θ 1 " 0.

Proposition. There is an Spp2θ, F p q-equivariant isomorphism X I θ 1 pw θ 1 q » Spp2θ, F q q{U K θ 1 ˆLK θ 1 X L K θ 1 I θ 1 pw θ 1 q, where X L K θ 1 I θ 1 pw θ 1 q is a Deligne-Lusztig variety for L K θ 1 . The zero-dimensional variety Spp2θ, F q q{U K θ 1 has a left action of Spp2θ, F q q and a right action of L K θ 1 .

Proof. It is similar to [START_REF] Muller | Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature (1,n-1)[END_REF] Proposition 8.

4.1.7

The Levi complement L K θ 1 is isomorphic to GLpθ ´θ1 q ˆSpp2θ 1 q, and its Weyl group is isomorphic to S θ´θ 1 ˆWθ 1 . Via this decomposition, the permutation w θ 1 corresponds to idˆw θ 1 . The Deligne-Lusztig variety X L K θ 1 I θ 1 pw θ 1 q decomposes as a product X L K θ 1 I θ 1 pw θ 1 q " X GLpθ´θ 1 q I θ 1 pidq ˆXSpp2θ 1 q H pw θ 1 q.

The variety X GLpθ´θ 1 q I θ 1 pidq is just a single point, but X Spp2θ 1 q H pw θ 1 q is the Coxeter variety for the symplectic group of size 2θ 1 . Indeed, w θ 1 is a Coxeter element, ie. the product of all the simple reflexions of the Weyl group of Spp2θ 1 q.

4.2 Unipotent representations of the finite symplectic group 4.2.1 Recall that a (complex) irreducible representation of a finite group of Lie type G " G F is said to be unipotent, if it occurs in the Deligne-Lusztig induction of the trivial representation of some maximal rational torus. Equivalently, it is unipotent if it occurs in the cohomology (with coefficient in Q ℓ with ℓ " p) of some Deligne-Lusztig variety of the form X B , with B a Borel subgroup of G containing a maximal rational torus. Let G, G 1 and let f : G Ñ G 1 be an F q -isotypy as in 4.1.2. If B is such a Borel in G, then B 1 :" f pBqZpG 1 q 0 is such a Borel in B 1 , and f induces an isomorphism X B " Ý Ñ X B 1 compatible with the actions. As a consequence, the map ρ Þ Ñ f ˝ρ defines a bijection between the sets of equivalence classes of unipotent representations of G 1 and of G. We will use this observation later in the case G " Spp2θq and G 1 " GSpp2θq, the symplectic group and the group of symplectic similitudes, the morphism f being the inclusion.

4.2.2

In this section, we recall the classification of the unipotent representations of the finite symplectic groups. The underlying combinatorics is described by Lusztig's notion of symbols. Our reference is [GM20] Section 4.4.

Definition. Let θ ě 1 and let d be an odd positive integer. The set of symbols of rank θ and defect d is Y 1 d,θ :" # S " pX, Y q ˇˇˇX " px 1 , . . . , x r`d q Y " py 1 , . . . , y r q with x i , y j P Z ě0 , x i`1 ´xi ě 1, y j`1 ´yj ě 1, rkpSq " θ

+ N pshiftq,
where the shift operation is defined by shiftpX, Y q :" pt0u \ pX `1q, t0u \ pY `1qq, and where the rank of S is given by rkpSq :" ÿ sPS s ´Z p#S ´1q 2 4 ^.

Note that the formula defining the rank is invariant under the shift operation, therefore it is well defined. By [START_REF] Lusztig | Irreducible representations of finite classical groups[END_REF], we have rkpSq ě

Y d 2 4
] so in particular Y 1 d,θ is empty for d big enough. We write Y 1 θ for the union of the Y 1 d,θ with d odd, this is a finite set. Example. In general, a symbol S " pX, Y q will be written S " ˜x1 . . . x r . . . x r`d y 1 . . . y r ¸.

We refer to X and Y as the first and second rows of S. The 6 elements of Y 1 2 are given by ˜2¸, ˜0 1 2

¸, ˜0 2 1 ¸, ˜1 2 0 ¸, ˜0 1 2 1 2 ¸, ˜0 1 2 ¸.
The last symbol has defect 3 whereas all the other symbols have defect 1.

4.2.3

The symbols can be used to classify the unipotent representations of the finite symplectic group.

Theorem ([Lus77] Theorem 8.2). There is a natural bijection between Y 1 θ and the set of equivalence classes of unipotent representations of Spp2θ, F q q. If S P Y 1 θ we write ρ S for the associated unipotent representation of Spp2θ, F q q. The classification is done so that the symbols ˜θ¸, ˜0 . . . θ ´1 θ 1 . . . θ

¸,

correspond respectively to the trivial and the Steinberg representations.

4.2.4

Let S " pX, Y q be a symbol and let k ě 1. A k-hook h in S is an integer z ě k such that z P X, z ´k R X or z P Y, z ´k R Y . A k-cohook c in S is an integer z ě k such that z P X, z ´k R Y or z P Y, z ´k R X. The integer k is referred to as the length of the hook h or the cohook c, it is denoted ℓphq or ℓpcq. The hook formula gives an expression of dimpρ S q in terms of hooks and cohooks.

Proposition ([GM20] Proposition 4.4.17). We have dimpρ S q " q apSq ś θ i"1 pq 2i ´1q 2 b 1 pSq ś h pq ℓphq ´1q ś c pq ℓpcq `1q

, where the products in the denominator run over all the hooks h and all the cohooks c in S, and the numbers apSq and b 1 pSq are given by apSq " ÿ Definition. The core of a symbol S P Y 1 d,θ is defined by corepSq :" S δ where d " 2δ `1. We say that S is cuspidal if S " corepSq.

Remark. In general, we have rkpcorepSqq ď rkpSq with equality if and only if S is cuspidal.

The next theorem states that cuspidal unipotent representations correspond to cuspidal symbols.

Theorem ([GM20] Theorem 4.4.28). The group Spp2θ, F q q admits a cuspidal unipotent representation if and only if θ " δpδ `1q for some δ ě 0. When this is the case, the cuspidal unipotent representation is unique and given by ρ S δ .

Theorem. Let S " pX, Y q P Y 1 d,θ . We have

˚RG G 1 ρ S " ÿ S 1 ρ S 1
where S 1 runs over all the symbols in Y 1 d,θ 1 such that, for some a 1 , a 2 ě 0 with a " a 1 `a2 , S 1 is obtained from S by removing an a 1 -hook of leg length 0 to its first row and an a 2 -hook of leg length 0 to its second row.

4.3

The cohomology of the Coxeter variety for the symplectic group 4.3.1 In this section we compute the cohomology of Coxeter varieties of finite symplectic groups, in terms of the classification of the unipotent characters that we recalled in 4.2.3.

Notation. We write X k :" X H pcoxq for the Coxeter variety attached to the symplectic group Spp2k, F q q, and H ' c pX k q instead of H ' c pX k b F, Q ℓ q where ℓ " p.

We first recall known facts on the cohomology of X k from Lusztig's work.

Theorem ( [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF]). The following statements hold.

(1) The variety X k has dimension k and is affine. The cohomology group H i c pX k q is zero unless k ď i ď 2k.

(2) The Frobenius F acts in a semisimple manner on the cohomology of X k .

(3) The groups H 2k´1 c pX k q and H 2k c pX k q are irreducible as Spp2k, F q q-representations, and the latter is the trivial representation. The Frobenius F acts with eigenvalues respectively q k´1 and q k . (4) The group H k`i c pX k q for 0 ď i ď k ´2 is the direct sum of two eigenspaces of F , for the eigenvalues q i and ´qi`1 . Each eigenspace is an irreducible unipotent representation of Spp2k, F q q. (5) The sum À iě0 H i c pX k q is multiplicity-free as a representation of Spp2k, F q q.

In other words, there exists a uniquely determined family of pairwise distinct symbols S k 0 , . . . , S k k and T k 0 , . . . , T k k´2 in Y 1 k such that

@0 ď i ď k ´2, H k`i c pX k q » ρ S k i ' ρ T k i , @k ´1 ď i ď k, H k`1 c pX k q » ρ S k i .
The representation ρ S k i (resp. ρ T k i ) corresponds to the eigenspace of the Frobenius F on À iě0 H i c pX k q attached to p i (resp. to ´pi`1 ). Moreover, we know that ρ S k k is the trivial representation, therefore

S k k " ˜k¸.
Lusztig also gives a formula computing the dimension of the eigenspaces. Specializing to the case of the symplectic group, it reduces to the following statement.

Proposition ([Lus76]). For 0 ď i ď k we have degpρ S k i q " q pk´iq 2 k´i ź s"1 q s`i ´1 q s ´1 k´i´1 ź s"0 q s`i `1 q s `1 .

For 0 ď j ď k ´2 we have degpρ T k j q " q pk´j´1q 2 pq k´1 ´1qpq k ´1q 2pq `1q k´j´2 ź

s"1 q s`j ´1 q s ´1 k´j´1 ź s"2 q s`j `1 q s `1 .

4.3.2 Our goal in this section is to determine the symbols S k i and T k j explicitly. This is done in the following proposition. We note that the statement is coherent with the two dimension formulae that we provided earlier. That is, the degree of ρ S k i (resp. of ρ T k j ) computed with the hook formula 4.2.4, agrees with the dimension of the eigenspace of p i (resp. of ´pj`1 ) in the cohomology of X k as given in the previous paragraph.

Proof. We use induction on k ě 0. Since we already know that S k k is the symbol corresponding to the trivial representation, the proposition is proved for k " 0. Thus we may assume k ě 1. We consider the block diagonal Levi complement L » GLp1, F q q ˆSpp2pk ´1q, F q q, and we write ˚Rk k´1 for the composition of the Harish-Chandra restriction from Spp2k, F q q to L, with the usual restriction from L to Spp2pk ´1q, F q q. As in the proof of [START_REF] Muller | Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature (1,n-1)[END_REF] Proposition 19, for all 0 ď i ď k we have an Spp2pk ´1q, F q q ˆxF y-equivariant isomorphism ˚Rk k´1

`Hk`i c pX k q ˘» H k´1`i c pX k´1 q ' H k´1`pi´1q c pX k´1 qp1q. (˚)

Here, p1q denotes the Tate twist. This recursive formula is established by Lusztig in [START_REF] Lusztig | Coxeter orbits and eigenspaces of Frobenius[END_REF] Corollary 2.10. The right-hand side is known by induction hypothesis whereas the left-hand side can be computed using 4.2.8 Theorem. We establish the proposition by comparing the different eigenspaces of F on both sides. If S P Y 1 d,k is any symbol, the restriction ˚Rk k´1 ρ S is the sum of all the representations ρ S 1 where S 1 is obtained from S by removing a 1-hook from any of its rows. We distinguish different cases depending on the values of k and i.

-Case k " 1. We only need to determine S 1 0 . For i " 0, the right-hand side of (˚) is ρ S 0 0 with eigenvalue 1. Thus, the symbol S 1 0 P Y 1 1 has defect 1 and admits only one 1-hook. If we remove this hook we obtain S 0 0 . Therefore, S 1 0 must be one of the two following symbols ˜0 1 1 ¸, ˜1¸.

By 4.3.1, we know that ρ S 1 0 has degree q, thus S 1 0 must be equal to the former symbol.

From now, we assume k ě 2 and we determine S k i for 0 ď i ă k. -Case k " 2 and i " 0. The eigenspace attached to 1 on the right-hand side of (˚) is ρ S 1 0 . Thus, the symbol S 2 0 P Y 1 k has defect 1 and admits only one 1-hook. If we remove this hook we obtain S 1 0 . Therefore, S 2 0 must be one of the two following symbols ˜0 1 2 1 2 ¸, ˜0 1 2 ¸.

By 4.3.1, we know that ρ S 2 0 has degree q 4 , thus S 2 0 must be equal to the former symbol. -Case k ą 2 and i " 0. The eigenspace attached to 1 on the right-hand side of (˚) is ρ S k´1 0 . Thus, the symbol S k 0 P Y 1 k has defect 1 and admits only one 1-hook. If we remove this hook we obtain S k´1 0 . The only such symbol is

S k 0 " ˜0 . . . k ´1 k 1 . . . k ¸.
-Case 1 ď i ď k ´1. The eigenspace attached to p i on the right-hand side of (˚) is

ρ S k´1 i ' ρ S k´1 i´1
. Thus, the symbol S k i P Y 1 k has defect 1 and admits only two 1-hooks. If we remove one of these hooks we obtain either S k´1 i or S k´1 i´1 . The only such symbol is

S k i " ˜0 . . . k ´i ´1 k 1 . . . k ´i ¸.
It remains to determine T k j for 0 ď j ď k ´2. -Case k " 2. The eigenspace attached to ´p on the right-hand side of (˚) is 0. Thus, the symbol T 2 0 P Y 1 2 has no hook at all, implying that it is cuspidal in the sense of 4.2.5. Since Spp4, F q q admits only 1 unipotent cuspidal representation, we deduce that

T 2 0 " ˜0 1 2 ¸.
-Case k " 3. First when j " 0, the eigenspace attached to ´p on the right-hand side of (˚) is ρ T 2 0 . Thus, the symbol T 3 0 P Y 1 3 has defect 3 and admits only one 1-hook. If we remove this hook we obtain T 2 0 . Therefore, T 3 0 must be one of the two following symbols ˜0 1 2 3 1 ¸, ˜0 1 3 ¸.

By 4.3.1, we know that ρ T 3 0 has degree q 4 pq 2 ´1qpq 3 ´1q 2pq`1q

, thus T 3 0 must be equal to the former symbol. Then when j " 1, the eigenspace attached to ´p2 on the right-hand side of (˚) is ρ T 2 0 . Thus, the symbol T 3 1 P Y 1 3 has defect 3 and admits only one 1-hook. If we remove this hook we obtain T 2 0 . Thus T 3 1 is also one of the two symbols above. We can deduce that it is equal to the latter by comparing the dimensions or by using the fact that the symbols T k j are pairwise distinct.

From now, we assume k ě 4 and we determine T k j for 0 ď j ď k ´2. -Case k " 4 and j " 0. The eigenspace attached to ´p on the right-hand side of (˚) is ρ T 3 0 . Thus, the symbol T 4 0 P Y 1 k has defect 3 and admits only one 1-hook. If we remove this hook we obtain T 3 0 . Therefore, T 4 0 must be one of the two following symbols ˜0 1 2 3 4 1 2 ¸, ˜0 1 2 3 2 ¸.

By 4.3.1, we know that ρ T 4 0 has degree q 9 pq 3 ´1qpq 4 ´1q 2pq`1q

, thus T 4 0 must be equal to the former symbol.

-Case k ą 4 and j " 0. The eigenspace attached to ´p on the right-hand side of (˚) is ρ T k´1 0 . Thus, the symbol T k 0 P Y 1 k has defect 3 and admits only one 1-hook. If we remove this hook we obtain T k´1 0 . The only such symbol is

T k 0 " ˜0 . . . k ´3 k ´2 k ´1 k 1 . . . k ´2 ¸.
-Case k " 4 and j " k ´2. The eigenspace attached to ´p3 on the right-hand side of (˚) is ρ T 3 1 . Thus, the symbol T 4 2 P Y 1 k has defect 3 and admits only one 1-hook. If we remove this hook we obtain T 3 1 . Therefore, T 4 2 must be one of the two following symbols ˜0 1 4 ¸, ˜0 2 3 ¸.

By 4.3.1, we know that ρ T 4 2 has degree q pq 3 ´1qpq 4 ´1q 2pq`1q

, thus T 4 2 must be equal to the former symbol.

-Case k ą 4 and j " k ´2. The eigenspace attached to ´pk´1 on the right-hand side of (˚) is ρ T k´1 k´3

. Thus, the symbol T k k´2 P Y 1 k has defect 3 and admits only one 1-hook. If we remove this hook we obtain T k´1 k´3 . The only such symbol is

T k k´2 " ˜0 1 k ¸.
-Case 1 ď j ď k ´3. The eigenspace attached to ´pj`1 on the right-hand side of (˚) is ρ T k´1 j ' ρ T k´1 j´1

. Thus, the symbol T k j P Y 1 k has defect 3 and admits only two 1-hooks. If we remove one of these hooks we obtain either T k´1 j or T k´1 j´1 . The only such symbol is

T k j " ˜0 . . . k ´j ´3 k ´j ´2 k ´j ´1 k 1 . . . k ´j ´2 ¸.
4.4 On the cohomology of a closed Bruhat-Tits stratum 4.4.1 Recall from 4.1.4 the θ-dimensional normal projective variety S θ :" X I ps θ q defined over F q . It is equipped with an action of the finite symplectic group Spp2θ, F q q. We use the Now, recall that the strata X I θ 1 pw θ 1 q are related to Coxeter varieties for the finite symplectic group Spp2θ 1 , F q q. Using 4.1.7, the geometric isomorphism given in 4.1.6 Proposition induces an isomorphism on the cohomology

H ' c pX I θ 1 pw θ 1 qq » R Spp2θ,Fqq L K θ 1
1 b H ' c pX Spp2θ 1 q pw θ 1 qq, (˚˚)

where L K θ 1 denotes the block-diagonal Levi complement isomorphic to GLpθ´θ 1 , F q qˆSpp2θ 1 , F q q. The variety X Spp2θ 1 q pw θ 1 q is nothing but the Coxeter variety that we denoted by X k 1 in 4.3.1, and whose cohomology we have described. For 0 ď i ď θ 1 and 0 ď j ď θ 1 ´2, recall from 4.3.2 the symbols S θ 1 i and T θ 1 j . We define

R S i,θ 1 :" R Spp2θ,Fqq L K θ 1 1 b ρ S θ 1 i , R T j,θ 1 :" R Spp2θ,Fqq L K θ 1 1 b ρ T θ 1 j .
Then by (˚˚), we have

H θ 1 `i c pX I θ 1 pw θ 1 qq » R S i,θ 1 ' R T i,θ 1 @0 ď i ď θ 1 ´2, H θ 1 `i c pX I θ 1 pw θ 1 qq » R S i,θ 1 @θ 1 ´1 ď i ď θ 1 .
The cohomology groups of other degrees vanish. The representation R S i,θ 1 corresponds to the eigenvalue q i of F , whereas R T j,θ 1 corresponds to ´qj`1 .

Lemma. Let 0 ď θ 1 ď θ, 0 ď i ď θ 1 and 0 ď j ď θ 1 ´2.

-If i ă θ 1 , the representation R S i,θ 1 is the multiplicity-free sum of the unipotent representations ρ S where S P Y 1 1,θ runs over the following 4 distinct families of symbols (S1) ˜0 . . . θ 1 ´i ´2 θ 1 ´i ´1 θ 1 `d 1 . . . θ 1 ´i ´1 θ ´i ´d ¸@0 ď d ď θ ´θ1 , (S2) ˜0 . . . θ 1 ´i ´2 θ 1 ´i ´1 `d θ 1 1 . . . θ 1 ´i ´1 θ ´i ´d ¸@1 ď d ď minpi, θ ´θ1 q, (S Exc 1) ˜0 . . . θ 1 ´i ´1 θ 1 ´i θ 1 . . . θ 1 ´i θ 1 ´i `1 ¸if θ 1 " θ, (S Exc 2) ˜0 . . . θ 1 ´i ´1 θ ´i ´1 θ 1 `1 1 . . . θ 1 ´i θ 1 ´i `1 ¸if θ 1 " θ, θ ´1 and θ ď θ 1 `i `1.

-The representation R S θ 1 ,θ 1 is the multiplicity-free sum of the unipotent representations ρ S where S P Y 1 1,θ runs over the following 2 distinct families of symbols (S1') ˜0 θ 1 `1 `d θ ´θ1 ´d ¸@0 ď d ď θ ´θ1 , (S2') ˜d θ 1 `1 θ ´θ1 ´d ¸@1 ď d ď minpθ 1 , θ ´θ1 q.

-If j `2 ă θ 1 , the representation R T j,θ 1 is the multiplicity-free sum of the unipotent representations ρ T where T P Y 1 3,θ runs over the following 4 distinct families of symbols (T1) ˜0 . . . θ 1 ´j ´4 θ 1 ´j ´3 θ 1 ´j ´2 θ 1 ´j ´1 θ 1 `d 1 . . . θ 1 ´j ´3 θ ´j ´2 ´d ¸@0 ď d ď θ ´θ1 , (T2) ˜0 . . . θ 1 ´j ´4 θ 1 ´j ´3 θ 1 ´j ´2 θ 1 ´j ´1 `d θ 1 1 . . . θ 1 ´j ´3 θ ´j ´2 ´d ¸@1 ď d ď minpj, θ ´θ1 q, (T Exc 1) ˜0 . . . θ 1 ´j ´2 θ 1 ´j ´1 θ 1 ´j θ 1 . . . θ 1 ´j ´1 ¸if θ 1 " θ, (T Exc 2) ˜0 . . . θ 1 ´j ´2 θ 1 ´j ´1 θ ´j ´1 θ 1 `1 1 . . . θ 1 ´j ´1 ¸if θ 1 " θ, θ ´1 and θ ď θ 1 `j `1.

-The representation R T θ 1 ´2,θ 1 is the multiplicity-free sum of the unipotent representations ρ T where T P Y 1 3,θ runs over the following 2 distinct families of symbols (T1')

˜0 1 2 θ 1 `1 `d θ ´θ1 ´d ¸@0 ď d ď θ ´θ1 , ( T2')
˜0 1 2 `d θ 1 `1 θ ´θ1 ´d ¸@1 ď d ď minpθ 1 ´2, θ ´θ1 q.

This lemma results directly from the computational rule explained in 4.2.7. In concrete terms, an induction of the form R Spp2θ,Fqq

L K θ 1 1 b ρ S 1
is the sum of all the representations ρ S where S is obtained from S 1 by adding a hook of leg length 0 to both rows, whose lengths sum to θ ´θ1 . We illustrate the arguments by looking at a concrete example.

With θ " 6, θ 1 " 3 and i " 2 let us explain the computation of

R S 2,3 " R Spp12,Fqq L K 3 1 b ρ S 3 2 . Recall that S 3 2 " ˜0 3 1 ¸.
For 0 ď d ď θ ´θ1 " 3, we add a d-hook of leg length 0 to the first row of S 3 2 , and a p3 ´dq-hook of leg length 0 to its second row. We may always add the hooks to the last entries of each row. By doing so we obtain the representations corresponding to the family of symbols (S1): . . . 4.4.5 Now, we have an explicit description of the terms E a,b 1 in the first page of the spectral sequence (E). In the Figure 6, we draw the shape of the first page.

R S 2,θ´2 ' R T 2,θ´2 R S 2,θ´1 ' R T 2,θ´1 R S 2,θ ' R T 2,θ R S 1,1 R S 1,2 . . . R S 1,θ´2 ' R T 1,θ´2 R S 1,θ´1 ' R T 1,θ´1 R S 1,θ ' R T 1,θ R S 0,0 R S 0,1 R S 0,2 ' R T 0,2 . . . R S 0,θ´2 ' R T 0,θ´2 R S 0,θ´1 ' R T 0,θ´1 R S 0,θ ' R T 0,θ
First, since the Frobenius F acts with the eigenvalue q i (resp. ´qj`1 ) on the representations R S i,θ 1 (resp. R T j,θ 1 ), 4.4.1 Proposition as well as point p1q of 4.4.2 and 4.4.3 Theorems follow from the triangular shape of the spectral sequence. Point p2q also follows from 4.4.4 Lemma. Next, we notice that on the b-th row of the first page E 1 , the eigenvalues of F which occur are q b and ´qb`1 . In particular, the eigenvalues on different rows are all distinct. It follows that all the arrows in the deeper pages of the sequence are zero, therefore it degenerates on the second page. Moreover, the filtration induced by the spectral sequence on the abutment splits, so that H k c pS θ q is isomorphic to the direct sum of the terms E k´b,b 2 on the k-th diagonal of the second page.

We prove point p3q of 4.4.2 and 4.4.3 Theorems. By the shape of the spectral sequence, we see that

H 2θ c pS θ q " H 2θ c pS θ q q θ » R S θ,θ » ρ ˜θ¸, H 2θ´2 c pS θ q ´qθ´1 » R T θ´2,θ » ρ ˜0 1 θ ¸.
Moreover, by the spectral sequence we know that H 0 c pS θ q is a subspace of R S 0,0 , thus the Frobenius F acts like the identity. Since S θ is projective and irreducible, the cohomology group H 0 c pS θ q " H 0 pS θ q is trivial.

We now prove point p4q of 4.4.2 and 4.4.3 Theorems. Let 2 ď i `2 ď θ 1 ď θ ´1. By extracting the eigenvalue q i in the spectral sequence, we have a chain

  -Tits stratification in Rapoport-Zink spaces . . . . . . . . . . . . . . . . . 1.2 The case of inert or ramified PEL unitary Rapoport-Zink space of signature p1, n ´1q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Step (A): the cohomology of an individual closed Bruhat-Tits stratum . . . . . . 1.4 Step (B): on the cohomology of the inert Rapoport-Zink space at hyperspecial level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The cohomology of the supersingular locus of the associated Shimura variety at an inert prime for n " 3, 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Adapting the approach to the ramified case . . . . . . . . . . . . . . . . . . . . . 2 Cohomology of the Bruhat-Tits strata in the unramified unitary Rapoport-Zink space of signature p1, n ´1q 2.1 The generalized Deligne-Lusztig variety X I pidq . . . . . . . . . . . . . . . . . . . 2.2 Irreducible unipotent representations of the finite unitary group . . . . . . . . . 2.3 Computing Harish-Chandra induction of unipotent representations in the finite unitary group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The cohomology of the Coxeter variety for the unitary group . . . . . . . . . . . 2.5 The cohomology of the variety X I pidq . . . . . . . . . . . . . . . . . . . . . . . . 3 On the cohomology of the basic unramified PEL unitary Rapoport-Zink space of signature p1, n ´1q 3.1 The Bruhat-Tits stratification on the PEL unitary Rapoport-Zink space of signature p1, n ´1q . . . . . . . . . . . . . . . . . . . . . 3.1.1 The PEL unitary Rapoport-Zink space M of signature p1, n ´1q . . . . 3.1.2 The Bruhat-Tits stratification of the special fiber M red . . . . . . . . . . 3.1.3 On the maximal parahoric subgroups of J . . . . . . . . . . . . . . . . . 3.1.4 Counting the closed Bruhat-Tits strata . . . . . . . . . . . . . . . . . . . 3.2 The cohomology of a closed Bruhat-Tits stratum . . . . . . . . . . . . . . . . . 3.3 Shimura variety and p-adic uniformization of the basic stratum . . . . . . . . . . 3.4 The cohomology of the Rapoport-Zink space at maximal level . . . . . . . . . . Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q 3.4.1 The spectral sequence associated to an open cover of M an . . . . . . . . 3.4.2 Compactly induced representations and type theory . . . . . . . . . . . . 3.4.3 The case n " 3, 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The cohomology of the basic stratum of the Shimura variety for n " 3, 4 . . . . 3.5.1 The Hochschild-Serre spectral sequence induced by p-adic uniformization 3.5.2 The case n " 3, 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 On the cohomology of the ordinary locus when n " 3 . . . . . . . . . . . 4 On the cohomology of a closed Bruhat-Tits stratum in the ramified PEL unitary Rapoport-Zink space of signature p1, n ´1q 4.1 The closed Deligne-Lusztig variety isomorphic to a closed Bruhat-Tits stratum . 4.2 Unipotent representations of the finite symplectic group . . . . . . . . . . . . . . 4.3 The cohomology of the Coxeter variety for the symplectic group . . . . . . . . . 4.4 On the cohomology of a closed Bruhat-Tits stratum . . . . . . . . . . . . . . . . -Tits stratification in Rapoport-Zink spaces

2 ľ

 2 pιpπq `π | LiepXqq " 0 and if n ě 3, n ľ pιpπq ´π | LiepXqq " 0, Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

  Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q 3.1.4.2 In[START_REF] Vollaard | The supersingular locus of the Shimura variety of GU(1,n-1) II[END_REF], an explicit formula is given for νpd ´1, dq. The next proposition gives a formula to compute νpr, dq for general r and d.Proposition. Let d ě 0 and let P d

  Coxeter graph of the reductive group G is a union of subgraphs of type A m (for various m). Let | W be the set of isomorphism classes of irreducible representations of its Weyl group W. The action of the Frobenius F on W induces an action on | W, and we consider the fixed point set | W F . The following theorem classifies the irreducible unipotent representations of G.

Proposition.

  Let b " 4 when n " 3, and b " 6 when n " 4. We have E ´1,b 2 " 0. By 3.4.1.8 Proposition, we may use the alternating Čech spectral sequence to show that E ´1,b

  pX X Y q . 4.2.5 For δ ě 0, we define the symbolS δ :" ˜0 . . . 2δ ¸P Y 1 2δ`1,δpδ`1q .

Proposition.

  For 0 ď i ď k and 0 ď j ď k ´2, we haveS k i " ˜0 . . . k ´i ´1 k 1 . . . k ´i ¸, T k j " ˜0 . . . k ´j ´3 k ´j ´2 k ´j ´1 k 1 . . . k ´j ´2¸.

  Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

Figure 6 :

 6 Figure 6: The first page of the spectral sequence.

  ), we refer to[START_REF] Renard | Représentations des groupes réductifs p-adiques[END_REF] VI.3.2 Proposition. The result there is stated in the context of a p-adic group G with normal subgroup H " 0 G such that G{ 0 G » Z d for some d ě 0, but the same arguments work as verbatim in the generality of the lemma. Admissibility of the representations involved is assumed only in order to apply Schur's lemma, insuring for instance the existence of central characters of smooth irreducible representations. In particular, if G{K is at most countable for any open compact subgroup K of G, then it is not necessary to assume admissibility. Let us prove (3). Assume towards a contradiction that π |H » σ ' σ g 0 and that σ » σ g 0 . We build a smooth admissible irreducible representation Π of G such that Π |H " σ, which results in a contradiction in regards to (2) since Hom H pΠ, πq " 0 but Π |H fi π |H . Let χ be the central character of π. Then χ |ZpGqXH coincides with the central character of σ.

must have the same parity. With these notations, the irreducible unipotent representations belonging to the principal series are those with cuspidal support pL 0 , ρ 0 q if k is even and pL 1 , ρ 1 q is k is odd.

Acknowledgements

Proposition. Let K be a parahoric subgroup of G and let ρ be a non cuspidal irreducible representation of its maximal reductive quotient K. Then no irreducible subquotient of c ´Ind G K ρ is supercuspidal.

3.4.2.10

We go back to the context of the unitary similitude group J. We may now determine the inertial support of any irreducible subquotient of a representation of the form c ´Ind J Jm ρ with ρ inflated from a unipotent representation of GUpV 0 m q. In particular, all the terms E 0,b 1 are of this form according to 3.4.1.11 Corollary. More precisely, let λ be a partition of 2m `1 and let ∆ t be its 2-core (see 3.2.8). Thus 2m `1 " tpt`1q 2 `2e for some e ě 0. The integer tpt`1q 2

is odd, so it can be written as 2f `1 for some f ě 0, and we have m " f `e. Using the basis of V 0 m fixed in 3.1.2.8, we identify GUpV 0 m q with the matrix group GU 2m`1 pF p q. The cuspidal support of ρ λ is pL t , ρ t q according to 3.2.8. Let P t be the standard parabolic subgroup with Levi complement L t . By direct computation, one may check that the preimage of P t in J m is the parahoric subgroup J f,...,m :" J f X J f `1 X . . . X J m . Let L f be the Levi complement of J that is associated to the parahoric subgroup J f,...,m . Using the basis of V fixed in 3.1.1.4, let V f be the subspace of V generated by V an and by the vectors e ˘1, . . . , e ˘f . It is equipped with the restriction of the hermitian form of V. Then L f » GpUpV f q ˆU1 pQ p q e q. The group L f X J f,...,m is a maximal parahoric subgroup of L f , and ρ t can be inflated to it. In particular, the pair pL f X J f,...,m , ρ t q is a level-0 type in L f . Since we work with unitary groups over an unramified quadratic extension, L f X J f,...,m is also a maximal compact subgroup of L f . In particular, pL f X J f,...,m , ρ t q is a type for a singleton of the form rL f , τ f s L f . Then τ f has the form τ f " c ´Ind

where r ρ t is some smooth irreducible representation of N L f pL f X J f,...,m q containing ρ t upon restriction. It follows that if we inflate ρ t to J f,...,m then pJ f,...,m , ρ t q is a rL f , τ f s-type in J. Moreover the compactly induced representation c ´Ind J Jm ρ λ is a quotient of c ´Ind J J f,...,m ρ t . In particular, we reach the following conclusion.

Proposition. Let λ be a partition of 2m `1 with 2-core ∆ t . Write tpt`1q 2 " 2f `1 for some f ě 0. Any irreducible subquotient of c ´Ind J Jm ρ λ has inertial support rL f , τ f s.

In particular, if f ă m then none of these irreducible subquotients are supercuspidal.

then no irreducible supercuspidal representation can occur. Thus V ρ λ ,χ,0 " 0. On the other hand, assume now that f " m so that L f " J and ρ λ is equal to the cuspidal representation ρ ∆m . As seen in 3.1.3.3, we have N J pJ m q " ZpJqJ m unless n " 2 (thus m " 0) in Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q quotient isomorphic to J{ZpJqJ 1 , which is trivial if n is odd and Z{2Z is n is even. Thus, we have

the last line following from the same Proposition 3.4 as above, but applied to J 1 . In [START_REF] Fargues | Cohomologie des espaces de modules de groupes p-divisibles et correspondances de Langlands locales[END_REF] Lemme 4.4.12, it is explained that Ext i J 1 pπ 1 , π 2 q vanishes for any smooth representations π 1 , π 2 of J 1 as soon as i is greater than the semisimple rank of J, that is 1 in our case. Hence, Ext 2 J,1 p1, St J q " 0 and we have

In particular, the right-hand side is non zero, which proves that pSt J q |J 1 contains St J 1 . If n is odd so that ZpJqJ 1 " J, it follows that pSt J q |J 1 " St J 1 . If n is even, in virtue of point (3) of the lemma, it remains to justify that for any g P J we have St g J 1 » St J 1 . This follows from the following computation

Let us go back to the irreducible representation π of J with unramified central character.

Summing up the previous paragraphs, we have that π |J 1 contains St J 1 if and only if π » χ ¨St J for some character χ of J that is trivial on J 1 (and thus trivial on ZpJ ˝qJ 1 " J ˝by the unramifiedness of the central character), and

3.5.2.3

We may now compute the cohomology of the basic stratum. Recall the supercuspidal representation τ 1 of the Levi complement M 1 Ă J that we defined in ??. When n " 3 or 4, we actually have M 1 " J and

is a supercuspidal representation of J, where N J pJ 1 q " ZpJqJ 1 (see 3.1.3.3) and Ą ρ ∆ 2 is the inflation of ρ ∆ 2 to N J pJ 1 q " ZpJqJ 1 (see 3.1.3.3) obtained by letting the center act trivially. We use the same notations as in 3.5.1.5.

There is a GpA p f q ˆW -equivariant monomorphism

There is a GpA p f q ˆW -equivariant monomorphism à

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

4 On the cohomology of a closed Bruhat-Tits stratum in the ramified PEL unitary Rapoport-Zink space of signature p1, n ´1q

Notations

Throughout the chapter, we fix an integer n ě 1 and an odd prime number p. If k is a perfect field of characteristic p, we denote by σ : x Þ Ñ x p the Frobenius of Autpk{F p q. If q " p e is a power of p, we write F q for the field with q elements. We fix an algebraic closure F of F p .

4.1 The closed Deligne-Lusztig variety isomorphic to a closed Bruhat-Tits stratum 4.1.1 Let q be a power of p and let G be a connected reductive group over F, together with a split F q -structure given by a geometric Frobenius morphism F . For H any F -stable subgroup of G, we write H :" H F for its group of F q -rational points. Let pT, Bq be a pair consisting of a maximal F -stable torus T contained in an F -stable Borel subgroup B. Let pW, Sq be the associated Coxeter system, where W " N G pTq{T. Since the F q -structure on G is split, the Frobenius F acts trivially on W. For I Ă S, let P I , U I , L I be respectively the standard parabolic subgroup of type I, its unipotent radical and its unique Levi complement containing T. Let W I be the subgroup of W generated by I.

For P any parabolic subgroup of G, the associated generalized parabolic Deligne-Lusztig variety is X P :" tgP P G{P | g ´1F pgq P PF pPqu.

We say that the variety is classical (as opposed to generalized) when in addition the parabolic subgroup P contains an F -stable Levi complement. Note that P itself needs not be F -stable.

We may give an equivalent definition using the Coxeter system pW, Sq. For I Ă S, let I W I be the set of elements w P W which are I-reduced-I. For w P I W I , the associated generalized parabolic Deligne-Lusztig variety is

The variety X I pwq is classical when w ´1I w " I, and it is defined over F q . The dimension is given by dim X I pwq " lpwq where lpwq denotes the length of w with respect to S.

4.1.2 Let G and G 1 be two reductive connected group over F both equipped with an F qstructure. We denote by F and F 1 the respective Frobenius morphisms. Let f : G Ñ G 1 be an F q -isotypy, that is a homomorphism defined over F q whose kernel is contained in the center of G and whose image contains the derived subgroup of G 1 . Then, according to [START_REF] Digne | Parabolic Deligne-Lusztig varieties[END_REF] proof of Proposition 11.3.8, we have G 1 " f pGqZpG 1 q 0 , where ZpG 1 q 0 is the connected component of unity of the center of G 1 . Thus intersecting with f pGq defines a bijection between parabolic

4.2.6

The determination of the cuspidal unipotent representations leads to a description of the unipotent Harish-Chandra series.

Definition. Let δ ě 0 such that θ " δpδ `1q `a for some a ě 0. We write L δ » GLp1, F q q a ˆSpp2δpδ `1q, F q q for the block-diagonal Levi complement in Spp2θ, F q q, with one middle block of size 2δpδ `1q and other blocks of size 1. We write ρ δ :" p1q a b ρ S δ , which is a cuspidal representation of L δ .

Proposition ([GM20] Proposition 4.4.29). Let S P Y 1 θ,d . The cuspidal support of ρ S is pL δ , ρ δ q where d " 2δ `1.

In particular, the defect of the symbol S of rank θ classifies the unipotent Harish-Chandra series of Spp2θ, F p q. 4.2.7 As it will be needed later, we explain how to compute a Harish-Chandra induction of the form

Definition. Let S " pX, Y q P Y 1 d,θ and let h be a k-hook of S given by some integer z. Assume that z P X and z ´k R X (resp. z P Y and z ´k R Y ). The leg length of h is given by the number of integers s P X (resp. Y ) such that z ´k ă s ă z. Consider the symbol S 1 " pX 1 , Y 1 q obtained by deleting z and replacing it with z ´k in the same row. We say that S 1 is obtained from S by removing a k-hook, or equivalently that S is obtained from S 1 by adding a k-hook.

where S runs over all the symbols in Y 1 d,θ such that, for some a 1 , a 2 ě 0 with a " a 1 `a2 , S is obtained from S 1 by adding an a 1 -hook of leg length 0 to its first row and an a 2 -hook of leg length 0 to its second row.

This computation is a consequence of the Howlett-Lehrer comparison theorem [START_REF] Howlett | Representations of Generic Algebras and Finite Groups of Lie Type[END_REF] as well as the Pieri rule for Coxeter groups of type B, see [GP00] 6.1.9. We will use it in concrete examples in the following sections.

4.2.8

There is a similar rule to compute Harish-Chandra restrictions. Let 0 ď θ 1 ď θ and consider the embedding G 1 ãÑ L ãÑ G where G 1 " Spp2θ 1 , F q q, G " Spp2θ, F q q and L is the block diagonal Levi complement GLpa, F q q ˆSpp2θ 1 , F q q where a " θ ´θ1 . We write ˚RG G 1 for the composition of the Harish-Chandra restriction functor ˚RG L with the usual restriction from L to G 1 .

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q stratification of 4.1.5 Proposition to study its cohomology over Q ℓ . If λ is a scalar, we write H ' c pS θ q λ to denote the eigenspace of the Frobenius F associated to λ (we do not in principle assume the eigenspace to be non zero). We give a series of statements before proving all of them at once in the remaining of this section.

Proposition. The Frobenius F acts semi-simply on H ' c pS θ q. Its eigenvalues form a subset of tq i | 0 ď i ď θu Y t´q j`1 | 0 ď j ď θ ´2u.

4.4.2

In a first statement, we give our results regarding the eigenspaces attached to a scalar of the form q i for some i. Recall from 4.2.6 the cuspidal supports pL δ , ρ δ q for the finite symplectic group Spp2θ, F q q. Theorem. Let 0 ď i ď θ and θ 1 P Z.

(1) The eigenspace H θ 1 `i c pS θ q q i is zero when θ 1 ă i or θ 1 ą θ.

We now assume that 0 ď i ď θ 1 ď θ.

(2) All the irreducible representations of Spp2θ, F q q in the eigenspace H θ 1 `i c pS θ q q i belong to the unipotent principal series, ie. they have cuspidal support pL 0 , ρ 0 q.

(3) We have

The cokernel of this map consists of at most 4 irreducible representations of Spp2θ, F q q. (5) When i " θ 1 " θ, we have

(7) When θ 1 " 1 and i " 0, we have

We note that when θ 1 " θ, the formula of p4q does not say anything about the eigenspace H θ`i c pS θ q q i since the sums are empty. However, by p6q we understand that this eigenspace is either 0 either irreducible. We note also that the theorem does not give any information in the case i `1 " θ 1 , except when θ 1 " 1 and i " 0 which corresponds to p7q.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q 4.4.3 In a second statement, we give our results regarding the eigenspaces attached to a scalar of the form ´qj`1 for some j.

Theorem. Let 0 ď j ď θ ´2 and θ 1 P Z.

(1) The eigenspace H θ 1 `j c pS θ q ´qj`1 is zero when θ 1 ă j `2 or θ 1 ą θ.

We now assume that 2 ď j `2 ď θ 1 ď θ.

(2) All the irreducible representations of Spp2θ, F q q in the eigenspace H θ 1 `j c pS θ q ´qj`1 are unipotent with cuspidal support pL 1 , ρ 1 q.

(3) We have

The cokernel of this map consists of at most 4 irreducible representations of Spp2θ, F q q. (5) When j `2 " θ 1 " θ, we have

(6) When θ 1 " θ we have

We note that when θ 1 " θ, the formula of p4q does not say anything about the eigenspace H θ`j c pS θ q ´qj`1 since the sums are empty. However, by p6q we understand that this eigenspace is either 0 either irreducible. We note also that the theorem does not give any information in the case j `3 " θ 1 .

Remark. A cuspidal representation occurs in the cohomology of S θ only in the cases θ " 0 and θ " 2. When θ " 0 it corresponds to H 0 c pS 0 q which is trivial. When θ " 2 it corresponds to H 2 c pS 2 q ´q as described by p3q in the theorem above.

4.4.4

The remaining of this section is dedicated to proving the theorems stated above. Recall from 4.1.5 that we have a stratification S θ " Ů θ θ 1 "0 X I θ 1 pw θ 1 q. It induces a spectral sequence on the cohomology whose first page is given by

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

When d ď minpθ ´θ1 , iq " minp3, 2q " 2, we may also add the first hook to the penultimate entry of the first row. Note that since i ă θ 1 , the first row of S θ 1 i has at least 2 entries. By doing so, we obtain the representations corresponding to the family of symbols (S2):

Now, recall that symbols are equal up to shifts. Therefore, one may rewrite S 3 2 as

Written this way, we notice that a 1-hook can be added to the first entry of the second row, which is a 0. Then one must add to the first row a hook of length d " θ ´θ1 ´1 " 2. One may always add it to the last entry, which results in the first "exceptional" representation (S Exc 1). Moreover if d ď i, which is the case here, one may also add this hook to the penultimate entry of the first row, which leads to the second "exceptional" representation (S Exc 2):

The sum of the representations attached to all the 8 symbols written above is isomorphic to R S 2,3 .

We also explain in detail the special case i " θ. Thus we compute

to the trivial representation of Spp2θ 1 , F q q. In order to compute this induction, we shift the symbol S θ 1 θ 1 first:

For 0 ď d ď θ ´θ1 , we add a d-hook of leg length 0 to the first row and a pθ ´θ1 ´dq-hook of leg length 0 to the second row. We may always add the hooks to the last entries of each row. By doing so, we obtain the representations corresponding to the family of symbols pS1 1 q. Moreover when d ď minpθ 1 , θ ´θ1 q, we may also add the first hook to the 0 in the first row. It leads to the representations corresponding to the family of symbols pS2 1 q.

In particular, we notice that the symbol of pS1 1 q with d " θ ´θ1 corresponds to the trivial representation of Spp2θ, F q q.

Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q

The quotient Kerpvq{Impuq is isomorphic to the eigenspace H θ 1 `i c pS θ q q i . The middle term R S i,θ 1 is the sum of the representations ρ S where S runs over the families of symbols (S1), (S2), (S Exc 1) and (S Exc 2) as in 4.4.4 Lemma. All these symbols are written in their "reduced" form, meaning that they can not be written as the shift of another symbol. Let us look at the length of the second row of these symbols. If S belongs to (S1) or (S2), then the second row has length θ 1 ´i. If S belongs to (S Exc 1) or (S Exc 2), then the second row has length θ 1 ´i `1. We may do a similar analysis for the left term (resp. the right term) by replacing θ 1 with θ 1 ´1 (resp. θ 1 `1). In the left term R S i,θ 1 ´1, all the representations corresponding to the families (S1) and (S2) have second row of length θ 1 ´i ´1. No such representation occurs in the middle term, therefore they all automatically lie in the Kerpuq. Then, in the left term the representation corresponding to (S Exc 1) occurs since θ 1 ´1 " θ. We observe that it is equivalent to the representation ρ S occuring in R S i,θ 1 with S in the family (S1) and d " θ ´θ1 . Further, assume that θ ď θ 1 `i so that the representation corresponding to (S Exc 2) occurs in R S i,θ 1 ´1. Then we observe that it is equivalent to the representation ρ S occuring in R S i,θ 1 with S in the family (S2) and d " θ ´θ1 " minpi, θ ´θ1 q. Hence, it follows that Impuq consists of at most 2 irreducible subrepresentations of R S i,θ 1 , and they correspond to the symbols of (S1) and (S2) with d " θ ´θ1 . Next, all the subrepresentations ρ S of R S i,θ 1 with S in (S1) or (S2) belong to Kerpvq, since no component of R S i,θ 1 `1 correspond to a symbol whose second row has length θ 1 ´i. Since θ 1 " θ, the represensation corresponding to (S Exc 1) occurs in R S i,θ 1 . We observe that it is equivalent to the representation ρ S occuring in R S i,θ 1 `1 with S in the family (S1) and d " θ ´θ1 ´1. Assume that θ 1 ď θ ´2 and θ ď θ 1 `i `1, so that the representation corresponding to (S Exc 2) occurs in R S i,θ 1 . Then we observe that it is equivalent to the representation ρ S occuring in R S i,θ 1 `1 with S in the family (S2) and d " θ ´θ1 ´1 " minpi, θ ´θ1 ´1q. Therefore, it is not possible to tell whether the components of R S i,θ 1 corresponding to (S Exc 1) and (S Exc 2) are in Kerpvq or not. In all cases, we conclude that Kerpvq{Impuq contains at least all the representations corresponding to the symbols S in (S1) and (S2) with d ă θ ´θ1 . With this description we miss up to four irreducible representations, which correspond to (S1) and (S2) with d " θ ´θ1 , (S Exc 1) and (S Exc 2). This proves point (4) of 4.4.2 Theorem. The point (4) of 4.4.3 Theorem is proved by identical arguments.

We now prove point p5q of 4.4.2 and 4.4.3 Theorems. We consider i " θ 1 " θ. By extracting the eigenvalue q i in the spectral sequence, we have a chain

The kernel Kerpuq is isomorphic to the eigenspace H 2i c pS θ q q i . The left term R S i,i is the sum of the representations ρ S 1 where S 1 runs over the families of symbols (S1') and (S2'). We observe that the representation ρ S 1 with S 1 in (S1') corresponding to some 0 ď d 1 ď θ ´i ´1 is equivalent to the component ρ S of R S i,i`1 with S in (S1) corresponding to d " d 1 . Similarly, we observe Cohomology of DL varieties associated to PEL RZ spaces with signature p1, n ´1q that the representation ρ S 1 with S 1 in (S2') corresponding to some 1 ď d 1 ď minpi, θ ´i ´1q is equivalent to the component ρ S of R S i,i`1 with S in (S2) corresponding to d " d 1 . Therefore, the representation ρ S corresponding to S in (S1') with d 1 " θ ´i belongs to Kerpuq. This is no other than the trivial representation. Moreover, if minpi, θ ´i ´1q " minpi, θ ´iq, ie. if 2i ě θ, then the representation ρ S corresponding to S in (S2') with d 1 " θ ´i also belongs to Kerpuq. This proves point (5) of 4.4.2 Theorem. The point (5) of 4.4.3 Theorem is proved by identical arguments.

Points (6) of 4.4.2 and 4.4.3 Theorems follows easily from the shape of the spectral sequence. Indeed, it suffices to notice that all the terms R S i,θ and R T j,θ in the rightmost column of the sequence are irreducible. Thus, they may either vanish, either remain the same in the second page.

Lastly we prove point (7) of 4.4.2. Assume first that θ " 1. The 0-th row of the spectral sequence is given by

We have H 1 c pS 1 q » Cokerpuq. Since we already know that H 0 c pS 1 q » Kerpuq is the trivial representation of Spp2, F q q, we see that u must be surjective. Therefore H 1 c pS 1 q " 0.

Remark. The vanishing of H 1 c pS 1 q also follows directly from the fact that S 1 » P 1 .

Let us now assume θ ě 2. The first terms of the 0-th row of the spectral sequence are R S 0,0

We have H 1 c pS θ q " H 1 c pS θ q 1 » Kerpvq{Impuq. The middle term R S 0,1 is the sum of all the representations corresponding to the following symbols ˜0 1 θ

On the other hand, the left term R S 0,0 is the sum of all the representations corresponding to the following symbols ˜0 1 `d θ ´d ¸, @0 ď d ď θ.

Since we already know that H 0 c pS θ q » Kerpuq is the trivial representation of Spp2θ, F q q, we see that Impuq contains all the components of R S 0,1 associated to a symbol whose second row has length 1. Therefore, H 1 c pS θ q is either 0 either irreducible, depending on whether the remaining component ˜0 1 θ 1 2