Annett Püttmann (joint with Dmitri Akhiezer)
For a connected complex reductive group G a normal complex G-variety X is called spherical if a Borel subgroup B of G has an open orbit in X. For X affine this is the case if and only if the algebra of regular functions on X is a multiplicity free G-module.
Similarly, a normal complex space X equipped with an action of a connected compact Lie group K by holomorphic transformations is called spherical the holomorphic tangent space at one point of X is generated by a Borel subalgebra. A normal Stein space is spherical if and only if the algebra of holomorphic functions is a multiplicity free K-module.
Faraut and Thomas gave an interesting and simple geometric condition which implies that the algebra of holomorphic functions is multiplicity free, namely, there exists an antiholomorphic involution that maps any K-orbit in X onto itself.
If X is a Stein manifold on which a connected compact Lie group K acts by holomorphic transformations, then the converse is true, i.e., the geometric condition implies that X is spherical. In order to prove this fact we need to understand antiholomorphic involutions of Stein manifolds with K-action that are equivariant with respect to a Weyl involution of K.