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The Harish-Chandra correlation functions, i.e. integrals over compact groupsof invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ...with the weight exp tr{X Omega Y Omega^dagger} are computed for the orthogonaland symplectic groups. We proceed in two steps. First, the integral over thecompact group is recast into a Gaussian integral over strictly upper triangularcomplex matrices (with some additional symmetries), supplemented by a summationover the Weyl group. This result follows from the study of loop equations in anassociated two-matrix integral and may be viewed as the adequate version ofDuistermaat-Heckman's theorem for our correlation function integrals. Secondly,the Gaussian integration over triangular matrices is carried out and leads tocompact determinantal expressions.Comment: 58 pages; Acknowledgements added; small corrections in appendix A;  minor changes & Note Adde

Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups