Resumen de A weighted Plancherel formula III: the case of the hyperbolic matrix ball

Jaak Peetre, Genkai Zhang

• The group SU(2,2) acts naturally on an $\L^2$-space on a hyperbolic matrix ball (type one bounded symmetric domain) with respect to the usual weighted measure. We will find the corresponding invariant Laplace operator and study its spectral resolution. The spherical functions ($K$-invariant eigenfunctions) can be expressed using hypergeometric functions. It turns out that, besides the weighted Bergman space, some discrete parts enter into the decomposition. The number of the discrete parts equals to the number of the orbits of the Weyl group action on the zeros (in the “lower half plane”) of the generalized Harish-Chandra $c$-function. We calculate their reproducing kernels in a special case. As an application, we obtain decompositions of the tensor products of holomorphic discrete series representations. This improves an earlier result by J. Repka