Boundary orbit strata and faces of invariant cones and complex Olshanskii semigroups

TL;DR

This paper provides a detailed description of the boundary orbit strata of the minimal Olshanskii semigroup associated with an irreducible Hermitian symmetric domain, linking geometric and representation-theoretic structures.

Contribution

It characterizes the K-orbit strata of the minimal Olshanskii semigroup as K-equivariant fiber bundles related to faces of the invariant cone, extending classical boundary stratum theory.

Findings

Complete description of K-orbit strata as fiber bundles

Identification of strata with conjugacy classes of faces

Connection between boundary geometry and invariant cones

Abstract

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.