## 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

### Keywords:

complex symmetric spaces; Stein manifolds

Zbl 0770.32016
Full Text: