Complex manifolds whose skeletons are semisimple real Lie groups, and analytic discrete series of representations. (English. Russian original) Zbl 0444.22006

Gel’fand, I. M.; Gindikin, S. G.

Funct. Anal. Appl. 11, 258-265 (1978); translation from Funkts. Anal. Prilozh. 11, No. 4, 19-27 (1977).

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.

Reviewer: I. L. Kantor (M.R. 58, 11230)

MSC:

22E46	Semisimple Lie groups and their representations
32Q28	Stein manifolds

Keywords:

irreducible unitary representations; complex Lie group; Stein manifold

References:

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