Boundary orbit strata and faces of invariant cones and complex Ol’shanskiĭ semigroups

What is it about?

Let D=G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification G_C, and there exists a closed complex subsemigroup Gamma, the so-called minimal Ol'shanskiĭ semigroup, characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to the interior of Gamma. 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 Gamma as K-equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra.

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