Remarks on Gelfand pairs associated to non-type-I solvable Lie groups
type: Item
instance of: scholarly article
title: Remarks on Gelfand pairs associated to non-type-I solvable Lie groups
zbMATH Open document ID: 0917.22006
author: Katsuhiko Kikuchi
published in: Journal of Functional Analysis
publication date: 1999-04-11T00:00:00Z
review text: Suppose that \(S\) is a connected and simply connected compact Lie group acting on \(S\) as automorphisms. \((K;S)\) is called a Gelfand pair if the Banach \(*\)-algebra \(L^1_K(S)\) of all \(K\)-invariant integrable functions on \(S\) is a commutative algebra. The main result is: \((K;S)\) is a Gelfand pair if and only if there exists a sufficiently large family \(F\subset\widehat S\) of irreducible unitary representations of \(S\) such that for each \(\pi\in F\): (1) the stabilizer \(K_\pi\) in \(K\) is compact; (2) each irreducible projective representation of \(K_\pi\) appears at most once in the decomposition of \(W_\pi\), where \(\widehat S\) is the unitary dual.
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 43A20
zbMATH DE Number: 1250654
zbMATH Keywords: compact Lie group
zbMATH Keywords: Gelfand pair
zbMATH Keywords: irreducible unitary representations
reviewed by: Su Weiyi
MaRDI profile type: Publication
full work available at URL: jfan.1998.3350
OpenAlex ID: W2024894435
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DOI: 10.1006/JFAN.1998.3350
description: scientific article; zbMATH DE number 1250654