On the Separation of Spectra in the Analysis of Berezin Kernels

The problem of restricting a highest weight representation of the group $\operatorname{U}(p,q)$ to the subgroup $\operatorname{O}(p,q)$ is considered. This restriction has an intricate spectrum that contains representations of different types. We construct a decomposition of this representation into reducible representations each of which has a single-type spectrum.
Some integrals over classical groups are also calculated; these integrals generalize those of Hua.