Weighted Bergman spaces associated with causal symmetric spaces

Let H\G be a causal symmetric space sitting inside its complexification H-C\G(C). Then there exist certain G-invariant Stein subdomains Xi of H-C\G(C). The Haar measure on H-C\G(C) gives rise to a G-invariant measure on Xi. With respect to this measure one can define the Bergman space B-2(Xi) of square integrable holomorphic functions on Xi. The group G acts unitarily on the Hilbert space B-2(Xi) by left translations in the arguments. The main result of this paper is the Plancherel Theorem for B-2(Xi), i.e., the disintegration formula for the left regular representation into irreducibles.