Gel’fand, I. M.; Gindikin, S. G. 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. Reviewer: I. L. Kantor (M.R. 58, 11230) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 2 Documents MSC: 22E46 Semisimple Lie groups and their representations 32Q28 Stein manifolds Keywords:irreducible unitary representations; complex Lie group; Stein manifold PDF BibTeX XML Cite \textit{I. M. Gel'fand} and \textit{S. G. Gindikin}, Funct. Anal. Appl. 11, 258--265 (1978; Zbl 0444.22006); translation from Funkts. Anal. Prilozh. 11, No. 4, 19--27 (1977) Full Text: DOI References: [1] V. Bargmann, ”Irreducible unitary representations of the Lorentz group,” Ann. Math.,48, 568-640 (1947). · Zbl 0045.38801 · doi:10.2307/1969129 [2] Harish-Chandra, ”Representations of semisimple Lie groups. VI,” Am. J. Math.,78, 564-628 (1956). · Zbl 0072.01702 · doi:10.2307/2372674 [3] S. Lang, SL2 (R), Addison-Wesley (1975). [4] I. M. Gel’fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation Theory and Automorphic Functions, Saunders (1969). [5] S. Bochner, ”Group invariance of Cauchy’s formula in several variables,” Ann. Math.,45, 686-707 (1944). · Zbl 0060.24301 · doi:10.2307/1969297 [6] S. G. Gindikin, ”Analysis in homogeneous spaces,” Usp. Mat. Nauk,19, No. 4, 3-92 (1964). · Zbl 0144.08101 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.