Quasiregular representations of the infinite-dimensional Borel group
type: Item
instance of: scholarly article
title: Quasiregular representations of the infinite-dimensional Borel group
zbMATH Open document ID: 1058.22015
published in: Journal of Functional Analysis
publication date: 2005-02-22T00:00:00Z
review text: The notion of quasiregular or geometric representation is well known for locally compact groups [\textit{A. A. Kirillov}, Elements of the theory of representations, Grundlehren der Mathematischen Wissenschaften 220 (Springer, Berlin) (1976; Zbl 0342.22001); Introduction to the theory of representations and noncommutative harmonic analysis, Representation theory and noncommutative harmonic analysis I, Encycl. Math. Sci. 22 (Springer, Berlin) 1--156 (1994; Zbl 0805.43004); \textit{D. P. Zhelobenko} and \textit{A. I. Shtern}, Representations of Lie Groups (Nauka, Moscow) (1983; Zbl 0521.22006)]. In this paper, the authors construct an analog of the quasiregular representation for the solvable infinite-dimensional Borel group and present a criterion of irreducibility of the constructed representations. This construction uses G-quasi-invariant Gaussian measures on some G-spaces and extends the method used in [\textit{A. V. Kosyak}, Funkts. Anal. Prilozh. 37, 78--81 (2003; Zbl 1022.22020)] for the construction of the quasiregular representations as applied to the nilpotent infinite-dimensional group.
reviewed by: Kun Soo Chang
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 43A80
Mathematics Subject Classification ID: 58D20
zbMATH DE Number: 2136768
zbMATH Keywords: infinite-dimensional group
zbMATH Keywords: Borel group
zbMATH Keywords: regular representations
zbMATH Keywords: quasiregular(geometric) representations
zbMATH Keywords: irreducibility
zbMATH Keywords: quasi-invariant measures
zbMATH Keywords: Ergodic measures
zbMATH Keywords: Ismagilov conjecture
zbMATH Keywords: solvable group
author: S. Albeverio
author: A. V. Kosyak
MaRDI profile type: Publication
full work available at URL: j.jfa.2004.03.009
OpenAlex ID: W2059751845
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DOI: 10.1016/J.JFA.2004.03.009
description: scientific article; zbMATH DE number 2136768