Discrete group actions on Stein domains in complex Lie groups

Details Discrete group actions on Stein domains in complex Lie groups Discrete group actions on Stein domains in complex Lie groups

2000-04-05

arxiv.org/abs/math/0004025v2

Forum Math. 16 (1) (2004), 37--68

Mathematics

Representation Theory

26 pages, to appear in Forum Math

Scientific paper

This paper deals with the analytic continuation of holomorphic automorphic forms on a Lie group $G$. We prove that for any discrete subgroup $\Gamma$ of $G$ there always exists a non-trivial holomorphic automorphic form, i.e., there exists a $\Gamma$-spherical unitary highest weight representation of $G$. Holomorphic automorphic forms have the property that they analytically extend to holomorphic functions on a complex Ol'shanski\u\i{} semigroup $S\subeq G_\C$. As an application we prove that the bounded holomorphic functions on $\Gamma\bs S\subseteq \Gamma\bs G_\C$ separate the points.

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