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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

##### MSC:
 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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