Deformation of irreducible unitary representations of discrete series of Hermitian symmetric simple Lie groups in the class of pure pseudorepresentations
type: Item
instance of: scholarly article
title: Deformation of irreducible unitary representations of discrete series of Hermitian symmetric simple Lie groups in the class of pure pseudorepresentations
zbMATH Open document ID: 1052.22010
DOI: 10.1023/A:1023242801314
published in: Mathematical Notes
publication date: 2004-11-15T00:00:00Z
review text: The deformation of irreducible unitary representations (IUR) of a simple Hermitian symmetric Lie group \(G\) is studied. It is shown that for each IUR \(\sigma \) of \(G\) in some Hilbert space \(H\) there is a one parameter family \(t\to \pi _{t}^{\sigma }\), \(t\in R\), of pairwise nonequivalent pure irreducible unitary \(\varepsilon _{t}\)-pseudorepresentations of \(G\) in the same space \(H\) such that the following two conditions hold: (1) \(\varepsilon _{t}\to 0\) as \(t\to 0\); (2) the norm \(\| \pi _{t}^{\sigma }( g) -\sigma ( g) \| \to 0\) as \(t\to 0\) for each \(g\in G\). This result is applied to construct explicitly the deformation of irreducible unitary principal series representations of \(G=SU(1,1)\). The universal covering group \(\widetilde{G}\) is defined and its nontrivial pseudocharacter \(\chi \) is evaluated. Then, a pure pseudorepresentation of the quotient group of \(\widetilde{G}\) by its center which is, in particular, a pure pseudorepresentation of the group \(SU(1,1)\), is constructed. It is proven that such pseudorepresentations are pairwise nonequivalent. It is also suggested that the deformation of the discrete series representations of \(SU(1,1)\) can be constructed in a similar way.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 57S20
zbMATH DE Number: 2115589
zbMATH Keywords: Hermitian symmetric simple Lie group
zbMATH Keywords: discrete series representation
zbMATH Keywords: pseudorepresentation
zbMATH Keywords: pure pseudorepresentation
zbMATH Keywords: pseudocharacter
author: A. I. Shtern
reviewed by: Gheorghe Zet
MaRDI profile type: Publication
full work available at URL: a:1023242801314
OpenAlex ID: W3092110934
description: scientific article; zbMATH DE number 2115589