Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds

Abstract

Given a Stein manifold XC which is homogeneous under a complex reductive Lie group GC, i.e., a complexification GC/KC of a compact homogeneous space G/K. Consider a relatively compact domain D which is invariant w.r.t. the compact real form G of the complex reductive Lie group in the Stein manifold XC. We find a relation between the automorphism group of the invariant domain D and isometric group of the compact homogeneous space G/K. When the compact homogeneous space G/K is isotropy irreducible, or even more general, we obtain a rigidity property of the automorphism groups.