Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants (Q2762316)

The author considers the irreducible decomposition of the so called canonical representation on the Cartan domain of type I, namely the domain \(D\) of contractive complex \(p\times q\) matrices \(z\) considered as a symmetric space \(D=G/K\) with \(G=U(p, q)\) and \(K=U(p)\times U(q)\). The canonical representation of \(G\) is defined by the positive definite kernel \({|K(z, w)|^2}{K(z, z)^{-1}K(w,w)^{-1}}\) where \(K(z, w)=\det(1-z w^\ast)^{-\nu}\) is the reproducing kernel of the weighted Bergman space for \(\nu >p+q-1\). It is also equivalent to the tensor product representation of the weighted Bergman space with its conjugate considered as projective representations of \(G\). In the present paper the author gives a survey of the study of the irreducible decomposition of the canonical representation and related problems. A unitary intertwining operator realizing the equivalence of the canonical representation with the standard regular representation of \(G\) on \(L^2(D)\) is found.