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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

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On pro-$p$-Iwahori invariants of $R$-representations of reductive $p$-adic groups
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by N. Abe, G. Henniart and M.-F. Vignéras PDF
Represent. Theory 22 (2018), 119-159 Request permission

Abstract:

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
  • Email: abenori@math.sci.hokudai.ac.jp
  • 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
  • Email: Guy.Henniart@math.u-psud.fr
  • M.-F. Vignéras
  • Affiliation: Institut de Mathématiques de Jussieu, 175 rue du Chevaleret, Paris 75013 France
  • Email: vigneras@math.jussieu.fr
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: Represent. Theory 22 (2018), 119-159
  • MSC (2010): Primary 20C08; Secondary 11F70
  • DOI: https://doi.org/10.1090/ert/518
  • MathSciNet review: 3864023