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 ( X1ΩY 1ΩX2 · · · ) with the weight exp tr ( XΩY 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. ♣ Service de Physique Théorique de Saclay, CEA/DSM/SPhT CNRS/SPM/URA 2306, F-91191 Gif-sur-Yvette Cedex, France. ♠ LPTHE Tour 24-25 5ème étage, Université Pierre et Marie Curie–Paris6, CNRS UMR 7589, 4 Place Jussieu, F 75252 Paris Cedex 5, France 1 E-mail: [email protected] 2 E-mail: [email protected] 3 E-mail: [email protected] 4 E-mail: [email protected]