Matrix analogues of the beta function and Plancherel's formula for Berezin kernel representations

Ten series of matrix integrals (over non-compact Riemannian symmetric spaces) imitating the standard beta function are constructed. This is a broad generalization of Hua Loo Keng's integrals (1958) and Gindikin's B-integrals (1964). As a consequence Plancherel's formula for the Berezin kernel representations of classical groups is obtained in explicit form. Matrix models of non-compact Riemannian symmetric spaces are also discussed.