HILBERT SPACES OF TENSOR-VALUED HOLOMORPHIC FUNCTIONS ON THE UNIT BALL OF Cn

The expansion of reproducing kernels of Bergman spaces of holomorphic functions on a domain D in Cn is of considerable interests. If the domain admits an action of a compact group K, then naturally one would like to decompose the space of polynomials into irreducible subspaces of K and expand the reproducing kernels in terms of the reproducing kernels of the finite dimensional subspaces. In [5] Hua found the expansion of the Bergman reproducing kernels on classical bounded symmetric domains. In that case the domain is a Hermitian symmetric space D = G/K, and the Bergman space forms also a unitary representation of the group G. Faraut and Koranyi [3] have recently found the expansion of weighted Bergman reproducing kernels on a general bounded symmetric domain and gave several implications. Since the appearance of that paper there have been more interests in the problem of finding the expansions. In the previous papers [6] and [7] the authors studied the expansions of reproducing kernels for spaces of vector-valued holomorphic functions on classical matrix tube domains U(n, n)/(U(n) × U(n)), Sp(n,R)/U(n) and O∗(2n)/U(n) (for even integers n). The tangent spaces of the domains are