Canonical representations on the two-sheeted hyperboloid
type: Item
instance of: scholarly article
title: Canonical representations on the two-sheeted hyperboloid
zbMATH Open document ID: 1107.43009
DOI: 10.1016/S0019-3577(05)80042-4
published in: Indagationes Mathematicae. New Series
publication date: 2006-12-07T00:00:00Z
review text: One of the sources for getting canonical representations on symmetric spaces \({\mathcal L}=G/H\) consists of the following. We take some group \(\widetilde G\) (overgroup) containing \(G\), take series representations \(\widetilde R\) of \(\widetilde G\) induced by the character of some parabolic subgroup \(\widetilde P\) associated with \(G/H\), and restrict these representations to \(G\). In the paper under review the author carries out this program for a crucial example: the two-sheeted hyperboloid \(\mathcal L\) in \({\mathbb R}^3\). The hyperboloid \(\mathcal L\) can be identified with the Riemann sphere \(\overline{\mathbb C}\) without the equator \(S: | z| =1\). Canonical representations of the group \(G=\text{SU}(1,1)\) are defined as the restriction of spherical representations of the overgroup \(\widetilde G=\text{SL}(2,{\mathbb C})\). The author decomposes these canonical representations into irreducible constituents and decomposes the Berezin form considered on the corresponding spaces.
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 5078037
zbMATH Keywords: Hermitian symmetric spaces
zbMATH Keywords: canonical representations
zbMATH Keywords: Berezin form
zbMATH Keywords: boundary representations
author: V. F. Molchanov
reviewed by: Sergei S. Platonov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5078037