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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 Gh/Kh. We describe the boundary orbits and relate them to the boundary orbits of Gh/Kh. We relate the crown and the split-holomorphic crown of G/K to the crown Ξh of Gh/Kh. 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.

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Journal of Functional Analysis