Spherical representations and mixed symmetric spaces

Let G/H be a symmetric space admitting a G-invariant hyperbolic cone field. For each such cone field we construct a local tube domain Ξ containing G/H as a boundary component. The domain Ξ is an orbit of an Ol’shanskii type semi group Γ. We describe the structure of the group G and the domain Ξ. Furthermore we explore the correspondence between Γ-modules of holomorphic sections of line bundles over Ξ and spherical highest weight modules. © 1997 American Mathematical Society