Analytic extension of the holomorphic discrete series. (English) Zbl 0626.43008

Motivated by I. M. Gel’fand and S. G. Gindikin [Funkts. Anal. Prilozh. 11, No.4, 19-27 (1977; Zbl 0444.22006)] and a classical theorem of Paley and Wiener, which describes an \(L^ 2\)-function on \({\mathbb{R}}\) as a sum of boundary values of holomorphic functions in the two domains of \({\mathbb{C}}\setminus {\mathbb{R}}\), the author derives results about analytic extensions of matrix-coefficients of a holomorphic discrete series of a semisimple Lie group G to certain domains in the complexification \(G_{{\mathbb{C}}}.\)
The author classifies the G-orbits on \(G_{{\mathbb{C}}}/G\) and relates it to Wolf’s classification of the G-orbits on \(G_{{\mathbb{C}}}/B\), where B is a Borel subgroup. - One should mention that both classifications has been generalized to semisimple symmetric spaces G/H, cf. T. Oshima and T. Matsuki [J. Math. Soc. Japan 32, 392-414 (1980; Zbl 0451.53039)] and T. Matsuki [ibid. 31, 331-357 (1979; Zbl 0396.53025)].
Reviewer: M.Flensted-Jensen


43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
42B25 Maximal functions, Littlewood-Paley theory
22E46 Semisimple Lie groups and their representations
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