Causal compactification and Hardy spaces for spaces of Hermitian type

Let G/H be a compactly causal symmetric space with causal compactification Phi : G/H -->S-1, where S-1 is the Bergman-Silov boundary of a tube type do main G(1)/K-1. The Hardy space H-2(C) of G/H is the space of holomorphic fu nctions on a domain Xi (C degrees) subset of G(C)/H-C with L-2-boundary val ues on G/H. We extend Phi to imbed Xi (C degrees) into G(1)/K-1, such that Xi (C degrees) = {z is an element of G(1)/K-1 \ psi (m)(z) not equal 0}, wi th psi (m) explicitly known. We use this to construct an isometry I of the classical Hardy space H-cl on G(1)/K-1 into H-2 (C) or into a Hardy space ( H) over tilde (2)(C) defined on a covering <(<Xi>)over tilde>(C degrees) of Xi (C degrees). we describe the image of I in terms of the highest weight modulus occurring in the decomposition of the Hardy space.