Complex manifolds whose skeletons are semisimple real Lie groups, and analytic discrete series of representations
type: Item
instance of: scholarly article
title: Complex manifolds whose skeletons are semisimple real Lie groups, and analytic discrete series of representations
zbMATH Open document ID: 0444.22006
DOI: 10.1007/BF01077140
publication date: 1978-01-01T00:00:00Z
review text: The authors propose a new approach to the study of irreducible unitary representations of a real semisimple Lie group \(G\). The basis of this approach consists of the following assertions: (1) the space of the complex Lie group \(G_c\) is partitioned into nonintersecting domains \(G_j^{(m)}\); more precisely, \(G_c=\cup\, \overline{G_j^{(m)}}\); (2) the group \(G\times G\) acts on each domain \(G_j^{(m)}\): \(g\to g_1^{-1}gg_2\), \(g_1,g_2\in G\), \(g\in G_c\); (3) the group \(G\) lies on the boundary of the domain \(G_j^{(m)}\) and is the skeleton. (The parameter \(j\) is a class of equivalent Cartan subgroups of \(G\), \((m)\) is the number of the Weyl chamber in a fixed Cartan subgroup of the given class.) In future publications the authors plan to study representations in function spaces on \(G_j^{(m)}\). In the present paper they consider the case when one of the domains \(G_j^{(m)}\) is a Stein manifold (this is possible if and only if \(G/U\), where \(U\) is a maximal compact subgroup, possesses a Hermitian symmetric structure). In the second part of the paper the case \(G=\mathrm{SL}(2,\mathbb R)\) is considered in detail.
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 32Q28
zbMATH DE Number: 3693584
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: complex Lie group
zbMATH Keywords: Stein manifold
Wikidata QID: Q115394339
author: I. M. Gel'fand
author: S. G. Gindikin
MaRDI profile type: Publication
cites work: Irreducible unitary representations of the Lorentz group
cites work: Representations of Semisimple Lie Groups VI: Integrable and Square-Integrable Representations
cites work: Group invariance of Cauchy's formula in several variables
cites work: Q5520117
full work available at URL: https://doi.org/10.1007/bf01077140
OpenAlex ID: W1974871597
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3693584