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Types and inductions for modular representations of \(p\)-adic groups (with an Appendix by M.-F. Vignéras). (Types et inductions pour les représentations modulaires des groupes \(p\)-adiques (avec un Appendice de M.-F. Vignéras).) (French) Zbl 0935.22013

Let \(G\) be the group of \(F\)-rational points of a connected reductive algebraic group defined over a non-Archimedean local field \(F\) of residue characteristic \(p\). Let \(\ell\) be a prime number such that \(\ell\neq p\). Let \((J_L,\tau_L)\) be a Bushnell-Kutzko type attached to a supercuspidal irreducible smooth complex representation \(\rho\) of a Levi subgroup \(L\) of a parabolic subgroup \(P\) of \(G\). Let \((J,\tau)\) be a cover [in the sense of C. Bushnell and P. Kutzko, Proc. Lond. Math. Soc., III. Ser. 77, No. 3, 582-634 (1998; Zbl 0911.22014)] of \((J_L,\tau_L)\). It follows from the theory of types by Bushnell and Kutzko that the category \(\mathfrak R_\tau(G)\) of smooth complex representations of \(G\) generated by their \(\tau\)-isotypic part is equivalent with the abelian category \(\mathfrak R_{\mathfrak s}(G)\) generated by the smooth complex irreducible representations of \(G\) with supercuspidal support in the inertia class \(\mathfrak s\) of \((L,\rho)\), and that the functor \(\mathfrak M_\tau\) from the category \(\mathfrak R_\tau(G)\) to the category of \({\mathcal H}(G//J,\tau)\)-modules, where \(\mathcal H(G//J,\tau):=\text{End}_G(\text{c-Ind}_J^G\tau)\) denotes the Hecke algebra of compactly supported \(\tilde\tau\)-spherical functions on \(G\), defined by \(\mathfrak M_\tau(\pi):=\operatorname{Hom}_J(\tau,\pi)\), is an equivalence of categories; here \(\text{c-Ind}\) denotes the smooth induction with compact support, and \(\tilde\tau\) denotes the contragrediant representation of \(\tau\).
That equivalence generalizes the usual equivalence between the category of complex smooth representations of \(G\) which are generated by their I-fixed vectors, where \(I\) is an Iwahori subgroup of \(G\), and the category of finite dimensional \({\mathcal H}(G//I,1)\)-modules, given by sending a representation \(V\) of \(G\) to the space \(V^I\) of its \(I\)-fixed vectors. Note that \({\mathcal H}(G//I,1)\) is simply the Iwahori-Hecke algebra of \(G\) with respect to \(I\), that is, the convolution algebra of compactly supported functions on \(G\) which are \(I\)-invariant on both sides. Let \(\overline{\mathbf F}_\ell\) denote an algebraic closure of a finite field with \(\ell\) elements. In the paper under review, the author uses the (\(\ell\)-modular version of) Bushnell-Kutzko theory of types to study the category \({\mathcal R}_ {\overline{F}_\ell}(G)\) of smooth \(\overline{\mathbf F}_\ell\)-representations of \(G\). He is able to compare the induction functor \(\text{c-Ind}_J^G\) and the normalized parabolic induction functor \(\text{i}_{L\subset P}^G\), though the above properties do not hold for \(\overline{\mathbf F}_\ell\)–representations (especially the category \(\mathfrak R_\tau(G)\) is not abelian). Under the assumptions that the representation \(\text{c-Ind}_J^G(\tau)\) is quasi-projective of finite type and that the algebra \(\mathcal H(G//J,\tau)\) satisfies some properties (usually satisfied for known types), he proves the existence of a natural isomorphism \(\Phi_{L,P}^G\colon\text{c-Ind}_J^G(\tau) \simeq (\text{i}_{L\subset P}^G\circ\text{c-Ind}_{J_L}^L)(\tau_L)\).
The paper starts with a description of “cuspidal types” in \(G\): they are types \((J,\tau)\) in \(G\) such that the (equivalence classes of) irreducible subquotients of \(\text{c-Ind}_J^G(\tau)\) are exactly the inertial class of a smooth irreducible cuspidal \(\overline{\mathbf F}_ \ell\)-representation of \(G\). Let \(\overline{P}\) denote the opposite of \(P\) relative to \(L\). Let \(\tau_L\) be a smooth irreducible \(\overline{\mathbf F}_\ell\)-representation of a compact open subgroup \(J_L\) of \(L\), and \((J,\tau)\) be a \(\overline{P}\)-cover of \((J_L ,\tau_L)\). There is a natural (up to renormalization) embedding of unital \(\overline{\mathbf F}_\ell\)-algebras \(t_ {\overline{P}} \colon {\mathcal H}(L//J_L,\tau_L)\rightarrow {\mathcal H}(G//J,\tau)\) (Bushnell-Kutzko). The author defines an intertwining operator \(\Phi_ {L,P}^G \in \operatorname{Hom}_G (\text{c-Ind}_J^G(\tau), (\text{i}_{L\subset P}^ G \circ \text{c-Ind}_{J_L}^L)(\tau_L))\) which is \({\mathcal H}(L//J_L,\tau_ L)\)-equivariant for the action on \(\text{c-Ind}_J^G(\tau)\) given by \(t_{\overline{P}}\). (The action on \(\text{i}_P^G \circ \text{c-Ind}_ {J_L}^L(\tau_L)\) being induced by the natural one on \(\text{c-Ind}_{J_L}^L(\tau_L)\).)
The main part of this paper consists of the description of \(\Phi_{L,P}^G\). In particular, if \((J_L ,\tau_L)\) is a cuspidal type in \(L\) and \((J,\tau)\) is a cover of \((J_L ,\tau_L)\), the author gives some extra conditions ensuring that \(\Phi_{L,P}^G\) is bijective. From this he derives some results about left adjunction and generic irreducibility. Finally, for \(G=\text{GL}_N(F)\), using the work of Bushnell-Kutzko, he checks that all the extra conditions are satisfied. This implies (by some Vignéras arguments given in the appendix) that the category \({\mathcal R}_{\overline{\mathbf F}_\ell}(G)\) is Noetherian and that the parabolic induction functor preserves the property of being of finite type.
Reviewer: A.M.Aubert (Paris)

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 0911.22014
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References:

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