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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

The 2020 MCQ for Representation Theory is 0.7.

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Inverse Satake isomorphism and change of weight
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by N. Abe, F. Herzig and M. F. Vignéras PDF
Represent. Theory 26 (2022), 264-324 Request permission

Abstract:

Let $G$ be any connected reductive $p$-adic group. Let $K\subset G$ be any special parahoric subgroup and $V,V’$ be any two irreducible smooth $\overline {\mathbb {F}_p}[K]$-modules. The main goal of this article is to compute the image of the Hecke bimodule $\operatorname {End}_{\overline {\mathbb {F}_p}[K]}(c-Ind_K^G V, c-Ind_K^G V’)$ by the generalized Satake transform and to give an explicit formula for its inverse, using the pro-$p$ Iwahori Hecke algebra of $G$. This immediately implies the “change of weight theorem” in the proof of the classification of mod $p$ irreducible admissible representations of $G$ in terms of supersingular ones. A simpler proof of the change of weight theorem, not using the pro-$p$ Iwahori Hecke algebra or the Lusztig-Kato formula, is given when $G$ is split and in the appendix when $G$ is quasi-split, for almost all $K$.
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Additional Information
  • N. Abe
  • Affiliation: Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
  • MR Author ID: 858099
  • Email: abenori@ms.u-tokyo.ac.jp
  • F. Herzig
  • Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Room 6290, Toronto, Ontario M5S 2E4, Canada
  • MR Author ID: 876324
  • Email: herzig@math.toronto.edu
  • M. F. Vignéras
  • Affiliation: Institut de Mathématiques de Jussieu, 4 place Jussieu, 75005 Paris, France
  • ORCID: 0000-0003-4442-3809
  • Email: vigneras@math.jussieu.fr
  • Received by editor(s): September 15, 2018
  • Received by editor(s) in revised form: September 24, 2021
  • Published electronically: March 21, 2022
  • Additional Notes: The first-named author was supported by JSPS KAKENHI Grant Number 18H01107. The second-named author was partially supported by a Sloan Fellowship, a Simons Fellowship, and an NSERC grant
  • © Copyright 2022 American Mathematical Society
  • Journal: Represent. Theory 26 (2022), 264-324
  • MSC (2020): Primary 20C08; Secondary 11F70
  • DOI: https://doi.org/10.1090/ert/594
  • MathSciNet review: 4397148