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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Modulo $ p$ representations of reductive $ p$-adic groups: Functorial properties

Authors: N. Abe, G. Henniart and M.-F. Vignéras
Journal: Trans. Amer. Math. Soc. 371 (2019), 8297-8337
MSC (2010): Primary 20C08; Secondary 11F70
Published electronically: March 25, 2019
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Abstract: Let $ F$ be a local field with residue characteristic $ p$, let $ C$ be an algebraically closed field of characteristic $ p$, and let $ \mathbf G$ be a connected reductive $ F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $ C$-representations of $ G=\mathbf G(F)$ in terms of supercuspidal representations of Levi subgroups of $ G$. Here, for a parabolic subgroup $ P$ of $ G$ with Levi subgroup $ M$ and an irreducible admissible $ C$-representation $ \tau $ of $ M$, we determine the lattice of subrepresentations of $ \textup {Ind}_P^G \tau $ and we show that $ \textup {Ind}_P^G \chi \tau $ is irreducible for a general unramified character $ \chi $ of $ M$. In the reverse direction, we compute the image by the two adjoints of $ \textup {Ind}_P^G$ of an irreducible admissible representation $ \pi $ of $ G$. On the way, we prove that the right adjoint of $ \textup {Ind}_P^G $ respects admissibility, hence coincides with Emerton's ordinary part functor $ \textup {Ord}_{\overline P}^G$ on admissible representations.

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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: Laboratoire de Mathématiques d’Orsay, Univ Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France

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

Received by editor(s): May 3, 2017
Received by editor(s) in revised form: August 28, 2017, and September 12, 2017
Published electronically: March 25, 2019
Additional Notes: The first-named author was supported by JSPS KAKENHI Grant Number 26707001.
Article copyright: © Copyright 2019 American Mathematical Society