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Discrete group actions on Stein domains in complex Lie groups. (English) Zbl 1044.22007

The authors study certain non-compact Stein manifolds \(S\) on which a real Lie group \(G\) acts freely but not transitively and for a discrete subgroup \(\Gamma\) of \(G\) address the question, whether \(\Gamma\setminus S\) is a Stein manifold. They succeed to answer the question for a special case, where \(\Gamma\) is the modular group. On the other hand they show that for cocompact \(\Gamma\) the bounded holomorphic functions separate the points of \(\Gamma\setminus S\).
A key fact in their argument is that \(S\) is a semigroup with a polar decomposition of the type \(S= G\times M\) and that a large part of the holomorphic discrete series of \(G\) admits a holomorphic extension to \(S\). This allows them to construct Poincaré series which are \(\Gamma\)-invariant (hyperfunction) vector for certain holomorphic representations.

MSC:

22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
22E40 Discrete subgroups of Lie groups
11F70 Representation-theoretic methods; automorphic representations over local and global fields
32D05 Domains of holomorphy
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