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Invariant domains in complex symmetric spaces. (English) Zbl 0803.32019

This paper investigates \(K\)-invariant domains in “complex symmetric spaces”, that is in Stein manifolds of the form \(K^ \mathbb{C}/L^ \mathbb{C}\), where \(K\) and \(L \subset K\) are compact Lie groups, forming a compact symmetric pair. The main results can be summarized in the following theorems.
Theorem 1. Let \(\Omega \Subset K^ \mathbb{C}/L^ \mathbb{C}\) be a \(K\)-invariant Stein domain in a complex symmetric space. Then
(a) \(\Omega\) contains a minimal orbit of type \(K/L\);
(b) \(\operatorname{Aut} (\Omega)\) stabilizes a minimal \(K\)-orbit of type \(K/L\);
(c) \(\operatorname{Aut} (\Omega)\) is a compact group.
Theorem 2. Let \(\Omega_ 1\), \(\Omega_ 2 \Subset K^ \mathbb{C}/L^ \mathbb{C}\) be \(K\)-invariant Stein domains in a complex symmetric space. Then \(\Omega_ 1\) and \(\Omega_ 2\) are biholomorphic if and only if there exists \(F \in \operatorname{Aut} (K^ \mathbb{C}/L^ \mathbb{C})\) such that \(F(\Omega_ 1) = \Omega_ 2\).
These theorems can be considered as natural generalization of certain results recently proved by P. Heinzner [Indiana Univ. Math. J. 41, No. 3, 707-712 (1992; Zbl 0770.32016)] for bounded domains in \(\mathbb{C}^ n\) containing the origin and invariant under a linear action by a compact group, without invariant holomorphic functions.
Reviewer: G.Fels

MSC:

32M05 Complex Lie groups, group actions on complex spaces
22E10 General properties and structure of complex Lie groups
32E10 Stein spaces

Citations:

Zbl 0770.32016
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