(Modified) Fredholm Determinants for Operators with Matrix-Valued Semi-Separable Integral Kernels Revisited

Let B be an irreducible hermitian symmetric domain of rank r in Cd and let G := Aut(B) the group of all biholomorphic automorphisms of B. We construct explicit integral formulas for the G-invariant inner products on spaces of holomorphic functions on B associated with the discrete Wallach points and the so-called embedded Wallach points, by means of integration on G-orbits on the boundary (theta)B of B and of invariant differential operators acting on symmetric domains of lower rank.