Correlation functions of Harish-Chandra integrals over the orthogonal and the symplectic groups

The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials ∏ tr(X^p1 Ω Y^q1 Ω^† X^p2 ....) with the weight exp∈tr∈(X Ω Y Ω ^† ) are computed for the orthogonal and symplectic groups. We proceed in two steps. First, the integral over the compact group is recast into a Gaussian integral over strictly upper triangular complex matrices (with some additional symmetries), supplemented by a summation over the Weyl group. This result follows from the study of loop equations in an associated two-matrix integral and may be viewed as the adequate version of Duistermaat-Heckman's theorem for our correlation function integrals. Secondly, the Gaussian integration over triangular matrices is carried out and leads to compact determinantal expressions.