Duality in the space of tempered virtual characters of a reductive \(p\)-group
type: Item
instance of: scholarly article
title: Duality in the space of tempered virtual characters of a reductive \(p\)-group
zbMATH Open document ID: 1121.22005
author: Karem Bettaïeb
published in: Manuscripta Mathematica
publication date: 2007-06-07T00:00:00Z
review text: Let \(G\) be a connected reductive group defined over a non-archimedean local field \(F\) of characteristic 0. The paper is concerned with a certain involution in the space of tempered virtual characters of \(G(F)\). The involution is given by the formula \[ D=\sum_{M}(-1)^{\text{dim}A_M}\, i^G_M \circ r^G_M, \] where \(i^G_M\) denotes parabolic induction and \(r^G_M\) parabolic restriction followed by taking the maximal tempered quotient. \(M\) runs through a set of standard Levi groups (in the paper \(M\) is supposed to run through a set of semi-standard Levi groups, but then the sum does not have the wanted properties). \(D\) commutes with induction and tempered restriction and \(D^2 = \text{id}\). From the symmetry of \(D\) (Lemma 4) it follows immediately that \(D\) maps each irreducible tempered character to \(\pm\) an irreducible tempered character (Theorem 5.1; the argument in the paper is obscure). The behaviour of \(D\) on elliptic characters is also treated.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 5162383
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
full work available at URL: s00229-006-0059-x
OpenAlex ID: W2049938385
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DOI: 10.1007/S00229-006-0059-X
description: scientific article; zbMATH DE number 5162383