The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels

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Consider a semisimple connected Lie group G G with an affine symmetric space X X . We study abstractly the intertwining operators from the discrete series of X X into representations with reproducing kernel and, in particular, into the discrete series of G G ; each such is given by a convolution with an analytic function. For X X of Hermitian type, we consider the holomorphic discrete series of X X and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question.