Spectral decompositions of some representations of Lie groups
type: Item
instance of: scholarly article
title: Spectral decompositions of some representations of Lie groups
zbMATH Open document ID: 0704.22013
DOI: 10.1007/BF01066374
author: N. Ya. Vilenkin
author: A. U. Klimyk
published in: Ukrainian Mathematical Journal
publication date: 1990-01-01T00:00:00Z
review text: Let G be one of the groups \(SO_ 0(p,q)\), U(p,q) and Sp(p,q). They are the translation groups of hyperbolic spaces. Let \(P=MAN\) be a maximal parabolic subgroup of G. Here A is a one-dimensional vector subgroup of G corresponding to a Cartan subspace of the hyperbolic space. The group M is a product \(M_ 1M_ 2\) of its subgroups where \(M_ 2\) is isomorphic to \({\mathbb{Z}}_ 2\), U(1) and Sp(1) respectively. Consider representations of G induced by representations \(m_ 1m_ 2an\mapsto a^{\sigma}\delta (m_ 2)\) of P where \(a\mapsto a^{\sigma}\), \(\sigma\in {\mathbb{C}}\), is a quasicharacter of A and \(\delta\in \hat M_ 2\). For these representations, the following things are considered in this note: the K-spectrum, the reducibility, the restrictions \(SO_ 0(2p,2q)\) on U(p,q), \(SO_ 0(4p,4q)\) and U(2p,2q) on Sp(p,q), and U(p,q) on U(1)\(\times U(p-1,q)\). Then the authors study some quasiregular representations of G. Earlier the decomposition of the quasiregular representation on the hyperbolic spaces was got by the reviewer, J. Faraut, W. Rossmann and others. The authors note that for complex and quaternion cases this decomposition can be got easily from the restrictions above and the decomposition for real case. The decompositions in this note have a rather qualitative character.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 4154763
zbMATH Keywords: translation groups
zbMATH Keywords: hyperbolic spaces
zbMATH Keywords: maximal parabolic subgroup
zbMATH Keywords: representations
zbMATH Keywords: reducibility
zbMATH Keywords: restrictions
zbMATH Keywords: quasiregular representations
zbMATH Keywords: decomposition
Wikidata QID: Q115394463
reviewed by: V. F. Molchanov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4154763