Reproducing Kernels and Composition Series for Spaces of Vector-Valued Holomorphic Functions on Tube Domains

We study Hilbert spaces of vector-valued holomorphic sections on tube domains of type II and III. We consider some small representations of the maximal compact subgroups and calculate the norm of each K-type in the Hilbert spaces. As a consequence we get composition series of the analytic continuations of certain holomorphic discrete series and expansions of matrix-valued reproducing kernels.