Some remarks on measures on non-compact semi-simple Lie groups
type: Item
instance of: scholarly article
title: Some remarks on measures on non-compact semi-simple Lie groups
zbMATH Open document ID: 0532.43001
author: Alladi Sitaram
published in: Pacific Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let G be an arbitrary connected, non-compact, semi-simple Lie group with finite centre, K a fixed maximal compact subgroup of G, and m a fixed Haar measure on G. The main results are the following three theorems. Theorem A: Let E be a right K-invariant Borel set in G such that \(0<m(E)<\infty\). If \(\mu\) is a complex (finite) right K-invariant measure on G such that \(\mu(gE)=0\) for all \(g\in G\), then \(\mu =0.\) Theorem B: Let E be a K-bi-invariant Borel set in G with \(\bar E\) compact and \(m(E)>0\). Then there exists an elementary spherical function \(\phi\) such that \(\int_{gE}\phi(x)dm(x)=0\) for all \(g\in G.\) Theorem C: Let G be the group SL(2,R) and E a K-bi-invariant Borel set in G with \(0<m(E)<\infty\). Then there exists a non-trivial function \(f\in L^ 1(G)\) such that \(\int_{gE}f(x)dm(x)=\int_{Eg}f(x)dm(x)=0\) for all \(g\in G\).
Mathematics Subject Classification ID: 43A05
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 28C10
zbMATH DE Number: 3844348
zbMATH Keywords: symmetric space of non-compact type
zbMATH Keywords: canonical G-invariant measure
zbMATH Keywords: invariant probability measure
zbMATH Keywords: measures on non-compact semi-simple Lie groups
reviewed by: Kokichi Sakai
MaRDI profile type: Publication
full work available at URL: pjm.1984.110.429
OpenAlex ID: W2028296148
Wikidata QID: Q115230778
DOI: 10.2140/PJM.1984.110.429
description: scientific article; zbMATH DE number 3844348