Restriction of discrete series of \(\text{SU}(2,1)\) to \(\text{S}(\text{U}(1)\times\text{U}(1,1))\)
type: Item
instance of: scholarly article
title: Restriction of discrete series of \(\text{SU}(2,1)\) to \(\text{S}(\text{U}(1)\times\text{U}(1,1))\)
zbMATH Open document ID: 0820.22009
author: Jiahai Xie
published in: Journal of Functional Analysis
publication date: 1995-08-29T00:00:00Z
review text: The group \(G = \text{SU}(2,1)\) consists of complex unimodular \(3 \times 3\) matrices preserving the Hermitian form \(| z_ 1|^ 2 + | z_ 2 |^ 2 - | z_ 3 |^ 2\). This group has several discrete series of irreducible unitary representations (holomorphic, antiholomorphic, and nonholomorphic). The author studies decompositions of restrictions of these representations onto the subgroup \(G_ 1 =\text{S}(\text{U}(1)\times\text{U}(1,1))\). These decompositions are multiplicity free. Both the discrete and the continuous parts enter the decompositions. To carry out concrete computations on \(G\), a realization for the discrete series on this group is described. The realization is based on the embedding of the discrete series into nonunitary principal series. The representation space for each discrete series of \(G\) becomes the completion of harmonic functions on \(S^ 3\) under suitable modified norm. The harmonic functions on \(S^ 3\) are treated in some detail. A particular basis, for each homogeneous harmonic function space, is chosen and labeled in terms of invariants for the action of \(G_ 1\). The action of the Lie algebra \({\mathfrak g}_ 1\) of \(G_ 1\) in this realization of discrete series representations then comes out explicitly as shifts along those homogeneous harmonic functions. The author uses the Casimir operator of \(G_ 1\) to determine decomposition of representations of \(G\) onto \(G_ 1\).
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 599810
zbMATH Keywords: spherical subgroups
zbMATH Keywords: Hermitian forms
zbMATH Keywords: discrete series of irreducible unitary representations
zbMATH Keywords: decompositions of restrictions
zbMATH Keywords: principal series
zbMATH Keywords: representation spaces
zbMATH Keywords: actions
zbMATH Keywords: discrete series representations
zbMATH Keywords: homogeneous harmonic functions
zbMATH Keywords: Casimir operators
reviewed by: A. U. Klimyk
MaRDI profile type: Publication
full work available at URL: jfan.1994.1077
OpenAlex ID: W1997862470
DOI: 10.1006/JFAN.1994.1077
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description: scientific article; zbMATH DE number 599810