Harmonic analysis on the infinite-dimensional unitary group
type: Item
instance of: scholarly article
title: Harmonic analysis on the infinite-dimensional unitary group
zbMATH Open document ID: 1255.43006
author: A. A. Osinenko
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-27T00:00:00Z
review text: The author considers the group \(U(\infty)\) whose elements are infinite unitary matrices \(U=\left[U_{ij}\right]\), \(i,j\in{\mathbb N}\), such that \(U_{ij}=\delta_{ij}\) for sufficiently large \(i+j\). The main goal of harmonic analysis on this group, as well as on another `large' group \(S(\infty)\), is to find a decomposition of the most `natural' unitary representations of this group into irreducible components. The article [\textit{G. Olshanski}, J. Funct. Anal. 205, No. 2, 464--524 (2003; Zbl 1036.43002)] contains a construction of a family \(T_{zw}\) of representations of \(U(\infty)\times U(\infty)\). The family is indexed by two complex parameters \(z\) and \(w\) satisfying the condition \(\Re(z+w)>-1/2\). As the author notes, the spectral type of \(T_{zw}\) is essentially dependent of whether or not \(z\) and \(w\) are integers. The case when both \(z\) and \(w\) are non-integer is already investigated in [\textit{A. Borodin} and \textit{G. Olshanski}, Ann. Math. (2) 161, No. 3, 1319--1422 (2005; Zbl 1082.43003)]. So, the present article is devoted to the case when both parameters are integer. As the author noticed, in this case, the spectrum of the decomposition changes dramatically. The representation \(T_{zw}\) is decomposed into a direct sum of subrepresentations \(T_{pqrs}\) which are called blocks. Then, the spectral type of \(T_{zw}\) is determined by a collection of measures on some finite dimensional faces \(\Omega(p,q;r,s)\). The result is similar to the one obtained in [\textit{S. Kerov, G. Olshanski} and \textit{A. Vershik}, Invent. Math. 158, No. 3, 551--642 (2004; Zbl 1057.43005)] for the group \(S(\infty)\) and corresponding one-parameter family of representations \(T_z\), for integer \(z\). The article is organized as follows. Chapter~1 contains the main definitions like characters of a topological group \(K\), and a summary of several useful results obtained in the area. Chapter~2 is devoted to the construction of \(T_{zw}\). Chapter~3 d…
Mathematics Subject Classification ID: 43A65
Mathematics Subject Classification ID: 22A10
zbMATH DE Number: 6110566
zbMATH Keywords: infinite-dimensional unitary group
zbMATH Keywords: unitary group representation
zbMATH Keywords: character of a topological group
zbMATH Keywords: disjointness of representations
zbMATH Keywords: Gelfand-Tsetlin graph
reviewed by: Alexander Tovstolis
MaRDI profile type: Publication
full work available at URL: s10958-012-0722-6
OpenAlex ID: W2058790216
cites work: Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes
cites work: Disjointness of representations arising in harmonic analysis on the infinite-dimensional unitary group
cites work: Multilateral Summation Theorems for Ordinary and Basic Hypergeometric Series in $U(n)$.
cites work: Harmonic analysis on the infinite symmetric group
cites work: The problem of harmonic analysis on the infinite-dimensional unitary group.
cites work: Représentations factorielles de type \(II_1\) de \(U(\infty)\)
DOI: 10.1007/S10958-012-0722-6
description: scientific article; zbMATH DE number 6110566