Extensions of real bounded symmetric domains

Abstract

For a real bounded symmetric domain, G/K, we construct various natural enlargements to which several aspects of harmonic analysis on G/K and G have extensions. Our starting point is the realization of G/K as a totally real submanifold in a bounded domain G h /K h . We describe the boundary orbits and relate them to the boundary orbits of G h /K h . We relate the crown and the split-holomorphic crown of G/K to the crown Ξ h of G h /K h . We identify an extension of a representation of K to a larger group Lc and use that to extend sections of vector bundles over the Borel compactification of G/K to its closure. Also, we show there is an analytic extension of K-finite matrix coefficients of G to a specific Matsuki cycle space.