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A weighted Plancherel formula. III: The case of the hyperbolic matrix ball. (English) Zbl 0836.43018
Summary: [Part I by {\it J. Peetre, L. Peng} and {\it G. Zhang}, A weighted Plancherel formula. The case of the unit disk. Application of Hankel operators. Technical report (Stockholm 1990); Part II by {\it G. Zhang} in Stud. Math. 102, 103-120 (1992; Zbl 0811.43003)]. 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 special 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 the number of the orbits of the Weyl group action on the zeros (in the “lower half plane”) of the generalized Harish-Chandra $\bold c$-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.

##### MSC:
 43A85 Analysis on homogeneous spaces 22E46 Semisimple Lie groups and their representations 43A65 Representations of groups, semigroups, etc. (abstract harmonic analysis) 43A90 Spherical functions (abstract harmonic analysis) 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (analytic spaces)
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