The holomorphic discrete series for affine symmetric spaces, I

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

“…It should be noted that in the literature such representations have been studied, e.g., by H. Schlichtkrull [17] by determining the Langlands-parameters of some of the representations in [4]. In [14] we found an explicit formula for 0 in the case of holomorphic line bundles over the symmetric space. In the general case we can only sayanalogously to the lowest K-type of the corresponding representation-that the function is given via the Flensted-Jensen-isomorphism and the Poisson transformation of a vector-valued distribution on the boundary of the dual noncompact Riemannian form Xr = G /K of X.…”

“…We collect some simple facts concerning intertwining operators and reproducing kernels in the first section of this paper. By this we get a general formula for the reproducing kernel in term of the "lowest" .K-type, and an abstract proof of the fact used in [14] that the inverse of an intertwining operator of a reproducing representation, i.e. a representation with a reproducing kernel, into L (X), is given by a convolution with a AT-finite, analytic function.…”

“…Thus Theorem 5.4 below gives the formula for the lowest /C-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.…”

“…In [14] we did this for the case of a holomorphic character of the universal covering of the complexification of a maximal compact subgroup. One of our main tools in that paper was the analysis of an intertwining operator into the holomorphic discrete series of G, and the fact that such an intertwining operator is determined by the reproducing kernel of the representation in its standard realization in the space of holomorphic functions on D. It turns out that this connection between the intertwining operators and the reproducing kernel only depends on the existence of a kernel.…”

Spherical unitary highest weight representations

“…Discrete series on G/H were constructed in [12] (see also [30]) and in [3] it was shown that (H −∞ ) H is one-dimensional for all discrete series on G/H except for four types of exceptional symmetric spaces. Holomorphic discrete series, i.e., unitary highest weight representation of G which can be realized in L 2 (G/H), were constructed in [28], [29] (this construction is essentially different from the one in [12]) and it was shown that (H −∞ ) H is always one-dimensional (this also covers half of the exceptional spaces in [3]). …”