Spherical Stein spaces

Let X be an irreducible reduced complex space on which a connected compact Lie group K acts by holomorphic automorphisms. Let G be the complexification of K and g the Lie algebra of G. Following the theory of algebraic transformation groups, we call the complex space X spherical if X is normal and its tangent space at some point is generated by the vector fields from a Borel subalgebra b or g. We give several characterizations of spherical Stein spaces. In particular, we prove that a connected Stein manifold X is spherical if and only if the algebra of K-invariant differential operators on X is commutative.