Function spaces and reproducing kernels on bounded symmetric domains

Abstract

There are several natural Hilbert spaces of holomorphic functions on a bounded symmetric domain. Such are the Bergman-type spaces, on which the holomorphic discrete series operates, and the Hardy-type spaces which are related to the analytic continuation of the holomorphic discrete series. Also closely related is the Bergmann space of entire functions on the ambient @” which arises as the closure of the polynomials with respect to a natural inner product. The space of holomorphic polynomials decomposes into irreducible subspaces under the action of the isotropy group K of the domain. The main facts about this decomposition were proved by Schmid [21]; for another proof see [22]. Each irreducible subspace contains a unique normalized L-invariant (“spherical”) polynomial, where L is the isotropy group of the Shilov boundary in K. Our first main result is the explicit computation of the norms of the spherical polynomials with respect to each of the Hilbert spaces considered. For the domains of classical type they were considered by Hua [9] and for some of the Hilbert spaces this was done before by Upmeier [24] using different methods; for certain others there are partial results in [22]. We are able to do this in a fairly simple unified way by making strong use of Gindikin’s generalized Gamma function [S]. Next we obtain a description of the reproducing kernels of the K-irreducible subspaces in each of our Hilbert spaces, and an expansion in terms of these for