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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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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

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

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