Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants

Consider the pseudounitary group $G=U(p,q)$ and its compact subgroup $K=U(p)\times U(q)$. We survey the analysis of the Berezin kernels on the symmetric space $G/K$. We also explicitly construct unitary intertwining operators from the Berezin representations of $G$ to the representation of $G$ in the space $L^2(G/K)$. This implies the existence of a canonical action of the group $G\times G$ in $L^2(G/K)$.