Modulo $p$ representations of reductive $p$-adic groups: Functorial properties
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- by N. Abe, G. Henniart and M.-F. Vignéras PDF
- Trans. Amer. Math. Soc. 371 (2019), 8297-8337 Request permission
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 $\mathrm {Ind}_P^G \tau$ and we show that $\mathrm {Ind}_P^G \chi \tau$ is irreducible for a general unramified character $\chi$ of $M$. In the reverse direction, we compute the image by the two adjoints of $\mathrm {Ind}_P^G$ of an irreducible admissible representation $\pi$ of $G$. On the way, we prove that the right adjoint of $\mathrm {Ind}_P^G$ respects admissibility, hence coincides with Emerton’s ordinary part functor $\mathrm {Ord}_{\overline P}^G$ on admissible representations.References
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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: Laboratoire de Mathématiques d’Orsay, Univ Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, 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): 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.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 8297-8337
- MSC (2010): Primary 20C08; Secondary 11F70
- DOI: https://doi.org/10.1090/tran/7406
- MathSciNet review: 3955548