\(L^2\)-singular dichotomy for orbital measures of classical compact Lie groups
type: Item
instance of: scholarly article
title: \(L^2\)-singular dichotomy for orbital measures of classical compact Lie groups
zbMATH Open document ID: 1179.43006
author: Sanjiv Gupta
author: Kathryn E. Hare
published in: Advances in Mathematics
publication date: 2009-10-14T00:00:00Z
review text: Let \(G\) be a compact Lie group. For any \(h\in G\) let \(C_h\) denote the conjugacy class containing \(h\). The orbital measure \(\mu_h\) supported on \(C_h\) is the \(G\)-invariant measure defined by \(\int_G f\, d\mu_h= \int_G f(ghg^{-1})\, dm_G(g)\) for \(f\) continuous, where \(m_G\) is the Haar measure on \(G\). In the paper under review the authors prove that for any classical, compact, simple, connected Lie group \(G\), the \(G\)-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising ``\(L^2\)-singular dichotomy'': Either \(\mu_h^k\in L^2(G)\) or \(\mu_h^k\) is singular to the Haar measure on \(G\), where \(\mu_h^k=\mu_h*\dots*\mu_h\) is the convolution power. The minimum exponent \(k\) for which \(\mu_h^k\in L^2(G)\) is specified; it depends on the Lie properties of the element \(h\in G\). As a corollary, the authors complete the solution to a classical problem -- to determine the minimum exponent \(k\) such that \(\mu^k\in L^1(G)\) for all central, continuous measures \(\mu\) on \(G\).
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 58C35
zbMATH DE Number: 5614813
zbMATH Keywords: compact Lie group
zbMATH Keywords: conjugacy class
zbMATH Keywords: orbital measure
zbMATH Keywords: singular measure
Wikidata QID: Q115362107
reviewed by: Sergei S. Platonov
MaRDI profile type: Publication
full work available at URL: j.aim.2009.06.008
OpenAlex ID: W1992614979
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DOI: 10.1016/J.AIM.2009.06.008
description: scientific article; zbMATH DE number 5614813