A commutative model of a representation of the group \(O(n, 1)^{X}\) and a generalized Lebesgue measure in the space of distributions
type: Item
instance of: scholarly article
title: A commutative model of a representation of the group \(O(n, 1)^{X}\) and a generalized Lebesgue measure in the space of distributions
zbMATH Open document ID: 1135.22018
publication date: 2007-06-05T00:00:00Z
review text: Let \(X\) be a smooth manifold and \(m\) a finite measure with a continuous density and \(G\) a Lie group. The paper under review continues a series of publications devoted to the representation theory of the corresponding current group \(G^X\) of bounded locally continuous mappings from \(X\) to \(G\) under pointwise multiplication. Here \(G=O(n,1)\) and the representation is what was called the basic representation in their earlier work. The authors describe a commutative model with respect to a natural maximal unipotent subgroup \(Z\) of \(G\). Then \(Z\) and the current group \(Z^X\) are both abelian, and the spectral theorem provides an \(L^2\)-space \(L^2(\nu)\) on which \(Z^X\) acts by multiplication group. If one realizes the representation of \(G^X\) on \(L^2(\nu)\), then one speaks about a commutative model. In this paper the representation on \(L^2(\nu)\) is a construction from scratch, and it is proved that the representation constructed is indeed the basic representation by relating it to the spherical functions coming from the basic representation.
reviewed by: Joachim Hilgert
Mathematics Subject Classification ID: 22E65
Mathematics Subject Classification ID: 22D10
Mathematics Subject Classification ID: 81R10
zbMATH DE Number: 5161600
zbMATH Keywords: Current groups
zbMATH Keywords: unitary representations
zbMATH Keywords: basic representation
zbMATH Keywords: maximal unipotent subgroup
author: M. I. Graev
author: A. M. Vershik
MaRDI profile type: Publication
full work available at URL: s10688-005-0021-9
OpenAlex ID: W1975433550
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DOI: 10.1007/S10688-005-0021-9
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 5161600