Hecke algebras with parameters and representations of a classical \(p\)-adic group: preservation of tempered spectrum
type: Item
instance of: scholarly article
title: Hecke algebras with parameters and representations of a classical \(p\)-adic group: preservation of tempered spectrum
zbMATH Open document ID: 1321.22022
author: Volker Heiermann
published in: Journal of Algebra
publication date: 2013-05-27T00:00:00Z
review text: Let \(G\) be a reductive \(p\)-adic group. Let \(M\) be a Levi subgroup of \(G\) and \(\sigma\) an irreducible cuspidal representation of \(M\). Fix an irreducible component \(\sigma_1\) of the restriction of \(\sigma\) to \(M^1\) and let \(\pi_1\) denote the parabolically induced representation \(i^G_P(\text{ind}^M_{M^1}(\sigma_1))\). Let \(\mathcal H\) denote the algebra \(\text{End}_G(\pi_1)\). In the Bernstein decomposition of \(Rep(G)\) the factor determined by \((M,\sigma)\) is equivalent to the category of all right \(\mathcal H\)-modules. The algebras \(\text{End}_G(\pi_1)\) have been computed explicitly by Heiermann when \(G\) is orthogonal or symplectic or an inner form of some \(\text{GL}_n\), cf. \textit{V. Heiermann} [Sel. Math., New Ser. 17, No. 3, 713--756 (2011; Zbl 1246.22021)] and [Int. Math. Res. Not. 2010, No. 9, 1607--1623 (2010; Zbl 1195.22012)]. In particular, \(\text{End}_G(\pi_1)\) is a semi-direct product of a Hecke algebra with parameters and the algebra of a finite group. In the present article, the author proves that the equivalence \(V\mapsto \text{Hom}_G(\pi_1,V)\) from \(Rep_{(M,\sigma)}(G)\) to the category of \(\mathcal H\)-modules preserves the tempered spectrum and the square integrable (mod centre of G) representations.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 20C08
zbMATH DE Number: 6166677
zbMATH Keywords: representations of \(p\)-adic groups
zbMATH Keywords: affine Hecke algebras
zbMATH Keywords: tempered representations
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2012.09.003
OpenAlex ID: W1997682724
cites work: Induced representations of reductive ${\germ p}$-adic groups. I
cites work: Unitary equivalences for reductive p-adic groups
cites work: Representations of Reductive p -Adic Groups: Localization of Hecke Algebras and Applications
cites work: The Schwartz algebra of an affine Hecke algebra
cites work: Hecke algebras and ${\rm SL}_n$-types
cites work: Intertwining operators and Hecke algebras with parameters from a reductive \(p\)-adic group: The case of the classical groups.
cites work: Parametres de Langlands et Algebres D'Entrelacement
cites work: DCOMPOSITION SPECTRALE ET REPRSENTATIONS SPCIALES DUN GROUPE RDUCTIF $p$-ADIQUE
cites work: Proof of the Deligne-Langlands conjecture for Hecke algebras
cites work: Affine Hecke Algebras and Their Graded Version
cites work: ON THE SPECTRAL DECOMPOSITION OF AFFINE HECKE ALGEBRAS
cites work: Q4797984
cites work: Discrete Series and Classification for p-Adic Groups I
cites work: LA FORMULE DE PLANCHEREL POUR LES GROUPES p-ADIQUES. DAPRS HARISH-CHANDRA
DOI: 10.1016/J.JALGEBRA.2012.09.003
description: scientific article; zbMATH DE number 6166677