Complex manifolds whose skeletons are semisimple real Lie groups, and analytic discrete series of representations (Q1144672)

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.