  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.
