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Consider a semisimple connected Lie group G with an aiSine symmetric space X. We study abstractly the intertwining operators from the discrete series of X into representations with reproducing kernel and in particular, into the discrete series of G; each such is given by a convolution with an analytic function. For X of Hermitian type, we consider the holomorphic discrete series of 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. 0. INTRODUCTION The purpose of this work is twofold. The first is to study abstractly the intertwining operators between discrete series representations of an affine symmetric space X and discrete series of the corresponding semisimple group. The second is to obtain very explicit formulas for certain discrete representations on X in effect giving the analogue of Harish-Chandra's original construction of the holomorphic discrete series in the group case. Not surprisingly, these two problems are highly connected. Thus Theorem 5.4 below gives the formula for the lowest K-type (the Flensted-Jensen function) on X, whereas in Theorem 7.3 essentially the same function gives the intertwining operator. Much of our analysis builds on the framework in [14], but otherwise we have tried to make the paper reasonably self-contained. This is, in particular, the case in §1, §2, and §4, where we abstract the properties of our intertwining operators with an eye towards later applications. We shall always let G be a connected, noncompact semisimple Lie group such that the associated Riemannian symmetric space is a bounded symmetric domain D c Cn . The best known and most interesting discrete series representations of G are the holomorphic discrete series representations. As explained above, one of our aims in this paper is to generalize this construction to affine symmetric spaces of Hermitian type and relate those with other well-known representations. Received by the editors September 23, 1988 and, in revised form, June 27, 1989. 1980 Mathematics Subject Classiffcation (1985 Rerision). Primary 22E46; Secondary 43A85. (D 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

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