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

Let D = G/K be an irreducible Hermitian symmetric domain. ThenG is contained in a complexificationGC, and there exists a closed complex subsemigroup G ⊂ Γ ⊂ GC, the so-called minimal Ol’shanskĭı semigroup, characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to Γ◦. Parallel to the classical theory of boundary strata for the symmetric domain D, due to Wolf and Korányi, 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.