Discrete group actions on Stein domains in complex Lie groups

Achab, Dehbia; Betten, Frank; Kroetz, Bernhard

Discrete group actions on Stein domains in complex Lie groups

Authors

Achab, Dehbia

Betten, Frank

Kroetz, Bernhard

Description

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.
26 pages, to appear in Forum Math