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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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