Resume Nous etudions les complexes de cohomologie equivariants des espaces symetriques p-adiques de Drinfeld munis de leur action de Galois, et nous montrons comment ils permettent de realiser la correspondance de Langlands locale pour certaines representations de GL d dites « principales elliptiques ». Il s'agit de la premiere etape d'une generalisation conjecturale de la theorie de Lubin–Tate non-abelienne de Carayol, qui fournirait une realisation cohomologique de la correspondance de Langlands pour toutes les representations elliptiques. Au cours de notre etude, nous calculons tous les groupes d'extensions entre representations elliptiques et les cup-produits correspondants, nous donnons un nouveau calcul de la cohomologie a supports compacts des espaces symetriques p-adiques, et nous obtenons une nouvelle preuve de la conjecture monodromie-poids de Deligne pour les varietes uniformisees par ces espaces.

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Espaces symétriques de Drinfeld et correspondance de Langlands locale

Smooth duality in natural characteristic

Let be a large field of characteristic . We classify the supersingular simple modules of the pro- -Iwahori Hecke -algebra of a general reductive -adic group . We show that the functor of pro- -Iwahori invariants behaves well when restricted to the representations compactly induced from an irreducible smooth -representation of a special parahoric subgroup of . We give an almost-isomorphism between the center of and the center of the spherical Hecke algebra , and a Satake-type isomorphism for . This generalizes results obtained by Ollivier for split and hyperspecial to general and special.

THE PRO-p-IWAHORI HECKE ALGEBRA OF A REDUCTIVE p-ADIC GROUP III (SPHERICAL HECKE ALGEBRAS AND SUPERSINGULAR MODULES)

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The pro- -Iwahori Hecke algebra of a reductive -adic group I

Let $$\mathrm{F}$$
 (resp. $$\mathbb F$$
 ) be a nonarchimedean locally compact field with residue characteristic p (resp. a finite field with characteristic p). For $$k=\mathrm{F}$$
 or $$k=\mathbb F$$
 , let $$\mathbf {G}$$
 be a connected reductive group over k and R be a commutative ring. We denote by $$\mathrm{Rep}( \mathbf G(k)) $$
 the category of smooth R-representations of $$ \mathbf G(k) $$
 . To a parabolic k-subgroup $${\mathbf P}=\mathbf {MN}$$
 of $$\mathbf G$$
 corresponds the parabolic induction functor $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}:\mathrm{Rep}( \mathbf M(k)) \rightarrow \mathrm{Rep}( \mathbf G(k))$$
 . This functor has a left and a right adjoint. Let $${{\mathcal {U}}}$$
 (resp. $${\mathbb {U}}$$
 ) be a pro-p Iwahori (resp. a p-Sylow) subgroup of $$ \mathbf G(k) $$
 compatible with $${\mathbf P}(k)$$
 when $$k=\mathrm{F}$$
 (resp. $$\mathbb F$$
 ). Let $${H_{ \mathbf G(k)}}$$
 denote the pro-p Iwahori (resp. unipotent) Hecke algebra of $$ \mathbf G(k) $$
 over R and $$\mathrm{Mod}({H_{ \mathbf G(k)}})$$
 the category of right modules over $${H_{ \mathbf G(k)}}$$
 . There is a functor $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}: \mathrm{Mod}({H_{ \mathbf M(k)}}) \rightarrow \mathrm{Mod}({H_{ \mathbf G(k) }})$$
 called parabolic induction for Hecke modules; it has a left and a right adjoint. We prove that the pro-p Iwahori (resp. unipotent) invariant functors commute with the parabolic induction functors, namely that $$\mathrm{Ind}_{\mathbf P(k)}^{\mathbf G(k)}$$
 and $$\mathrm{Ind}_{{H_{ \mathbf M(k)}}}^{{H_{ \mathbf G(k)}}}$$
 form a commutative diagram with the $${{\mathcal {U}}}$$
 and $${{\mathcal {U}}}\cap \mathbf M(\mathrm{F})$$
 (resp. $${\mathbb {U}}$$
 and $${\mathbb {U}}\cap \mathbf M(\mathbb F) $$
 ) invariant functors. We prove that the pro-p Iwahori (resp. unipotent) invariant functors also commute with the right adjoints of the parabolic induction functors. However, they do not commute with the left adjoints of the parabolic induction functors in general; they do if p is invertible in R. When R is an algebraically closed field of characteristic p, we show that an irreducible admissible R-representation of $$ \mathbf G(\mathrm{F}) $$
 is supercuspidal (or equivalently supersingular) if and only if the $${H_{ \mathbf G(\mathrm{F})}}$$
 -module $${\mathfrak {m}}$$
 of its $${{\mathcal {U}}}$$
 -invariants admits a supersingular subquotient, if and only if $${\mathfrak {m}}$$
 is supersingular.

Parabolic induction in characteristic p

Let F be a locally compact non-archimedean field, p its residue characteristic, and G a connected reductive group over F. Let C an algebraically closed field of characteristic p. We give a complete classification of irreducible admissible C-representations of G = G(F), in terms of supercuspidal C-representations of the Levi subgroups of G, and parabolic induction. Thus we push to their natural conclusion the ideas of the third-named author, who treated the case G = GL_m, as further expanded by the first-named author, who treated split groups G. As in the split case, we first get a classification in terms of supersingular representations of Levi subgroups, and as a consequence show that supersingularity is the same as supercuspidality.

A classification of irreducible admissible mod p representations of p-adic reductive groups

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$.

Modular representations of p-adic groups

Suppose that $\\mathbf{G}$ is a connected reductive group over a finite extension $F/\\mathbb{Q}_{p}$ and that $C$ is a field of characteristic $p$. We prove that the group $\\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.

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Marie-France Vigneras

Papers

A classification of irreducible admissible mod p representations of p-adic reductive groups

A classification of irreducible admissible mod p representations of p-adic reductive groups

Modular representations of p-adic groups

On the existence of admissible supersingular representations of -adic reductive groups