The holomorphic discrete series for affine symmetric spaces, I

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Let X = G H be an affine symmetric space. We study part of the discrete spectrum of L 2( X) for X of Hermitian type, a notion we define in analogy with the group case. In particular we find intertwining operators from the scalar holomorphic discrete series of G, which is automatically of Hermitian type, into L 2( X). The multiplicity of this series is then shown to be one by a uniqueness result for the intertwining operators. Finally, we investigate the complexification X C of X and show that the discrete series in question admits holomorphic continuation into a certain domain in X C .