Holomorphic H-spherical distribution vectors in principal series representations

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown Ξ. In this article we construct a minimal G-invariant subdomain ΞH of Ξ with G/H as Shilov boundary. Let π be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of π, which admit a holomorphic extension to ΞH, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L2(G/H)mc in a space of holomorphic functions.