Reduced \(C ^{*}\)-algebra of the \(p\)-adic group \(GL(n)\). II
type: Item
instance of: scholarly article
title: Reduced \(C ^{*}\)-algebra of the \(p\)-adic group \(GL(n)\). II
zbMATH Open document ID: 1014.22014
published in: Journal of Functional Analysis
publication date: 2003-03-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0110018
review text: Let \(F\) be a non-Archimedean local field, let \(G=GL(n;F),\) and let \(C^*_{\text{red}}(G)\) denote the reduced \(C^*\)-algebra of \(G.\) Let \({\mathcal H}(G)\) be the Hecke algebra of \(G,\) and let \(\Omega\) be a component in the Bernstein variety \(\Omega (G).\) The Bernstein decomposition \({\mathcal H}(G)=\oplus {\mathcal H}(\Omega)\) is a canonical decomposition of \({\mathcal H}(G)\) into two-sided ideals \({\mathcal H}(\Omega).\) A minimal refinement of this decomposition is given. Let \(C^*_{\text{red}}(G//I)\) be the reduced Iwahori-Hecke \(C^*\)-algebra of \(G.\) Structure theorems for this unital \(C^*\)-algebra and the reduced spherical \(C^*\)-algebra are proved using standard techniques of non-commutative Fourier analysis. These results lead to a very explicit description of the tempered dual of \(G=GL(n;F)\) in terms of Bernstein parameters and extended quotients. It it proved that the Plancherel measure on the tempered dual of a reductive \(p\)-adic group is rotation-invariant.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 46L10
zbMATH DE Number: 1888367
zbMATH Keywords: reduced \(C^*\)-algebra
zbMATH Keywords: Bernstein decomposition
zbMATH Keywords: non-commutative Fourier analysis
zbMATH Keywords: Plancherel measure
zbMATH Keywords: reductive \(p\)-adic group
author: Roger Plymen
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
OpenAlex ID: W2012540056
arXiv ID: math/0110018
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DOI: 10.1006/JFAN.2002.3980
description: scientific article