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Spherical representations and mixed symmetric spaces. (English) Zbl 0887.22022
Summary: 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 $$\Xi$$ containing $$G/H$$ as a boundary component. The domain $$\Xi$$ is an orbit of an Ol’shanskii type semigroup $$\Gamma$$. We describe the structure of the group $$G$$ and the domain $$\Xi$$. Furthermore we explore the correspondence between $$\Gamma$$-modules of holomorphic sections of line bundles over $$\Xi$$ and spherical highest weight modules.

##### MSC:
 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 22E15 General properties and structure of real Lie groups 53C35 Differential geometry of symmetric spaces 54H15 Transformation groups and semigroups (topological aspects)
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