Boundary Orbit Strata and Faces of Invariant Cones and Complex Ol’shanskĭı Semigroups

Let D = G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification GC, and there exists a closed complex subsemigroup G ⊂ Γ ⊂ GC characterised by fact that all holomorphic discrete series representations of G extend holomorphically to Γ◦, the socalled minimal Ol’shanskĭı semigroup. Parallel to the classical theory of boundary strata for the symmetric domain D, due to Korányi and Wolf, we give a detailed and complete description of the K-orbit type strata of Γ as K-equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra g.