Discrete group actions on Stein domains in complex Lie groups

Abstract

This paper deals with the analytic continuation of holomorphic automorphic forms on a Lie group $G$. We prove that for any discrete subgroup $Γ$ of $G$ there always exists a non-trivial holomorphic automorphic form, i.e., there exists a $Γ$-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\uı semigroup $S\subeq G_\C$. As an application we prove that the bounded holomorphic functions on $Γ\bs S\subseteq Γ\bs G_\C$ separate the points.