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

##### MSC:
 22E46 Semisimple Lie groups and their representations 32Q28 Stein manifolds
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##### References:
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