Lower multiplicity for irreducible representations of nilpotent locally compact groups
type: Item
instance of: scholarly article
title: Lower multiplicity for irreducible representations of nilpotent locally compact groups
zbMATH Open document ID: 0943.22004
published in: Journal of Functional Analysis
publication date: 2000-09-13T00:00:00Z
review text: The author studies the lower multiplicity \(M_L(\pi)\) for irreducible representations \(\pi\) of a nilpotent locally compact group \(G\). The first result says that if the nilpotent group \(G\) is connected, then for every irreducible representation \(\pi\) of \(G\), such that \(\pi\) is not open in \(\widehat{G}\), the lower multiplicity \(M_L(\pi)= 1\). For an arbitrary non compact nilpotent group \(G\), it is shown that \(M_L(\pi)= 1\) for every finite-dimensional irreducible representation \(\pi\). The next result concerns nilpotent SIN-groups. A locally compact group \(G\) is called a SIN-group if \(G\) has a neighbourhood basis of the identity consisting of sets \(V\) such that \(xVx^{-1}=V\) for all \(x\in G\). The author shows that for a noncompact amenable SIN-group \(G\) with primitive ideal space \(T_1\), the dual space \(\widehat{G}\) has no open points and so \(M_L(\pi)\) is defined for every irreducible representation \(\pi\). Moreover, if \(M_L(\pi)\) is finite, then \(\pi\) is finite-dimensional. As a corollary, it follows that for any noncompact nilpotent SIN-group \(G\) and for every \(\pi\in \widehat{G}\) the number \(M_L(\pi)\) is defined and either \(M_L(\pi)=1\) or \(M_L(\pi)= \infty\). Furthermore \(M_L(\pi)=1\) if and only if \(\pi\) is finite-dimensional.
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 43A90
zbMATH DE Number: 1389439
zbMATH Keywords: lower multiplicity
zbMATH Keywords: irreducible representations
zbMATH Keywords: nilpotent locally compact group
zbMATH Keywords: nilpotent SIN-groups
author: Eberhard Kaniuth
reviewed by: Jean Ludwig
MaRDI profile type: Publication
OpenAlex ID: W1999634544
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DOI: 10.1006/JFAN.1999.3469
description: scientific article; zbMATH DE number 1389439