Causal compactification and Hardy spaces

Let $\mathcal{M}=G/H$ be a irreducible symmetric space of Cayley type. Then $\mathcal{M}$ is diffeomorphic to an open and dense $G$-orbit in the Shilov boundary of $G/K\times G/K$. This compactification of $\mathcal{M}$ is causal and can be used to give answers to questions in harmonic analysis on $\mathcal{M}$. In particular we relate the Hardy space of $\mathcal{M}$ to the classical Hardy space on the bounded symmetric domain $G/K\times G/K$. This gives a new formula for the Cauchy-Szegö kernel for $\mathcal{M}$.