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A characterization of $$K$$-invariant Stein domains in symmetric embeddings. (English) Zbl 0790.32030
Ancona, Vincenzo (ed.) et al., Complex analysis and geometry. New York: Plenum Press. The University Series in Mathematics. 223-234 (1993).
Let $$K$$ be a connected compact Lie group and let $$G$$ be its complexification. Let $$X$$ be an irreducible normal Stein space equipped with a $$G$$-action. If $$G$$ has an open orbit in $$X$$ that is a complex symmetric space, then $$X$$ is called an affine symmetric embedding. Each such $$X$$ possesses an essentially canonical closed complex subvariety $$D$$ that is an affine torus embedding with respect to a certain subgroup of $$G$$ such that $$K\cdot D=X$$.
The main result of the paper is the following theorem: Let $$\Omega$$ be a connected $$K$$-invariant domain in an affine symmetric embedding $$X$$. Then $$\Omega$$ is Stein if and only if $$\Omega\cap D$$ is Stein and connected.
For the entire collection see [Zbl 0772.00007].

##### MSC:
 32M05 Complex Lie groups, group actions on complex spaces 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) 32E10 Stein spaces, Stein manifolds