  Consider the pseidounitary group $G=U(p,q)$ and its compact subgroup
$K=U(p)$. We construct an explicit unitary intertwining operator from the
tensor product of a holomorphic representation and a antiholomorphic
representation of $G$ to the space $L^2(G/K)$. This implies the existense of a
canonical action of the group $G\times G$ in $L^2(G/K)$. We also give a survey
of analysis of Berezin kernels and their relations with special functions.
