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