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. W ith respect to this measure one can define the Bergman space B-2(Xi) of squ are integrable holomorphic functions on Xi. The group G acts unitarily on t he Hilbert space B-2(Xi) by left translations in the arguments. The main re sult of this paper is the Plancherel Theorem for B-2(Xi), i.e., the disinte gration formula for the left regular representation into irreducibles.