Parabolic induction in characteristic p

Autor: Rachel Ollivier, Marie-France Vignéras
Rok vydání: 2018
Předmět:
Zdroj: Selecta Mathematica. 24:3973-4039
ISSN: 1420-9020
1022-1824
DOI: 10.1007/s00029-018-0440-0
Popis: 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.
Databáze: OpenAIRE
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