Parabolic induction and Hecke modules in characteristic \(p\) for \(p\)-adic GL\(_n\)
type: Item
instance of: scholarly article
title: Parabolic induction and Hecke modules in characteristic \(p\) for \(p\)-adic GL\(_n\)
zbMATH Open document ID: 1243.22017
author: Rachel Ollivier
published in: Algebra \& Number Theory
publication date: 2010-11-02T00:00:00Z
review text: Let \(F\) be a local non-archimedean field of residual characteristic \(p\), let \(k\) be an algebraic closure of \(\mathbb F_p\), and let \(G:=\text{GL}_n(F)\). The author studies \(k\)-representations of \(G\). A key tool is the functor \(V\mapsto V^{I(1)}\) that associates to a smooth representation \(V\) of \(G\) its subspace of invariant vectors under the pro-\(p\)-Iwahori subgroup \(I(1)\) of \(G\), that is, \(I(1)\) is the pro-\(p\)-radical of the standard Iwahori subgroup \(I\) of \(G\). The space \(V^{I(1)}\) is a module over the \textit{pro-\(p\)-Iwahori-Hecke algebra} \(\mathcal H=\mathcal H(G)\) of compactly supported functions \(f: I(1)\backslash G/I(1)\to\mathbb Z\). The structure of \(\mathcal H\) has been studied by Vignéras, \(\mathcal H\) a finite extension of an affine Hecke algebra \(\mathcal H_{\text{aff}}\) and it admits a Bernstein-type presentation with only positive powers of the cardinal of the residue field of \(F\). The author has obtained previously the classification of the simple modules of \(\mathcal H\) in the case \(n=3\). Three families of \(\mathcal H\)-modules appear: the \textit{regular}, \textit{singular}, and \textit{supersingular} ones. In the article under review the author proves in particular a conjecture stated by Vignéras in 2005, which says that any non-zero simple supersingular module contains a character of \(\mathcal H_{\text{aff}}\). It implies the following numerical Langlands correspondence on the side of \(\mathcal H\)-modules: The number of \(n\)-dimensional simple supersingular \(\mathcal H\)-modules (with fixed action of the uniformizer) is equal to the number of smooth irreducible \(n\)-dimensional mod \(p\) representations of the absolute Galois group of \(F\) (with fixed determinant of a Frobenius). On the other hand, the author investigates the non-supersingular \(\mathcal H\)-modules by defining a process of induction from the pro-\(p\)-Iwahori-Hecke algebra \(\mathcal H(L)\) of a Levi subgroup \(L\) of \…
reviewed by: Anne-Marie Aubert
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 20C08
Mathematics Subject Classification ID: 20G05
zbMATH DE Number: 5809196
zbMATH Keywords: mod p representation of Hecke algebras and p-adic groups
zbMATH Keywords: parabolic induction
zbMATH Keywords: integral Bernstein presentation
zbMATH Keywords: Satake transform
MaRDI profile type: Publication
full work available at URL: ant.2010.4.701
OpenAlex ID: W2070467432
DOI: 10.2140/ANT.2010.4.701
description: scientific article; zbMATH DE number 5809196