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On pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups

Authors: N. Abe, G. Henniart and M.-F. Vignéras
Journal: Represent. Theory 22 (2018), 119-159
MSC (2010): Primary 20C08; Secondary 11F70
Published electronically: October 15, 2018
MathSciNet review: 3864023
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Let $F$ be a locally compact field with residue characteristic $p$, and let $\mathbf {G}$ be a connected reductive $F$-group. Let $\mathcal {U}$ be a pro-$p$ Iwahori subgroup of $G = \mathbf {G}(F)$. Fix a commutative ring $R$. If $\pi$ is a smooth $R[G]$-representation, the space of invariants $\pi ^{\mathcal {U}}$ is a right module over the Hecke algebra $\mathcal {H}$ of $\mathcal {U}$ in $G$.

Let $P$ be a parabolic subgroup of $G$ with a Levi decomposition $P = MN$ adapted to $\mathcal {U}$. We complement a previous investigation of Ollivier-Vignéras on the relation between taking $\mathcal {U}$-invariants and various functor like $\operatorname {Ind}_P^G$ and right and left adjoints. More precisely the authors’ previous work with Herzig introduced representations $I_G(P,\sigma ,Q)$ where $\sigma$ is a smooth representation of $M$ extending, trivially on $N$, to a larger parabolic subgroup $P(\sigma )$, and $Q$ is a parabolic subgroup between $P$ and $P(\sigma )$. Here we relate $I_G(P,\sigma ,Q)^{\mathcal {U}}$ to an analogously defined $\mathcal {H}$-module $I_\mathcal {H}(P,\sigma ^{\mathcal {U}_M},Q)$, where $\mathcal {U}_M = \mathcal {U}\cap M$ and $\sigma ^{\mathcal {U}_M}$ is seen as a module over the Hecke algebra $\mathcal {H}_M$ of $\mathcal {U}_M$ in $M$. In the reverse direction, if $\mathcal {V}$ is a right $\mathcal {H}_M$-module, we relate $I_\mathcal {H}(P,\mathcal {V},Q)\otimes \operatorname {c-Ind}_\mathcal {U}^G\mathbf {1}$ to $I_G(P,\mathcal {V}\otimes _{\mathcal {H}_M}\operatorname {c-Ind}_{\mathcal {U}_M}^M\mathbf {1},Q)$. As an application we prove that if $R$ is an algebraically closed field of characteristic $p$, and $\pi$ is an irreducible admissible representation of $G$, then the contragredient of $\pi$ is $0$ unless $\pi$ has finite dimension.

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Additional Information

N. Abe
Affiliation: Department of Mathematics, Hokkaido University, Kita 10, Nishi 8, Kita-Ku, Sapporo, Hokkaido, 060-0810, Japan
MR Author ID: 858099

G. Henniart
Affiliation: Université de Paris-Sud, Laboratoire de Mathématiques d’Orsay, Orsay cedex F-91405 France; CNRS, Orsay cedex F-91405 France
MR Author ID: 84385

M.-F. Vignéras
Affiliation: Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, Paris 75013 France

Keywords: Parabolic induction, pro-$p$ Iwahori Hecke algebra
Received by editor(s): March 14, 2018
Received by editor(s) in revised form: June 17, 2018
Published electronically: October 15, 2018
Additional Notes: The first-named author was supported by JSPS KAKENHI Grant Number 26707001.
Article copyright: © Copyright 2018 American Mathematical Society