# Extensions of real bounded symmetric domains

## TL;DR

This paper explores natural enlargements of real bounded symmetric domains, extending harmonic analysis, boundary orbit relations, and representation theory to broader contexts.

## Contribution

It introduces new extensions of real bounded symmetric domains, connecting boundary orbits, crown domains, and representation extensions.

## Key findings

- Boundary orbits of G/K relate to those of G_h/K_h.
- Extensions of K-representations to larger groups are constructed.
- K-finite matrix coefficients extend analytically to Matsuki cycle spaces.

## 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 \Xi_h of G_h/K_h. We identify an extension of a representation of K to a larger group L_c 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.