Publication : t06/131

Abstract:
The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials $\prod \,\mathrm{tr}\>\left(X^{p_1} \Omega Y^{q_1}\Omega ^\dagger X^{p_2}\cdots\right)$ with the weight $\exp \,\mathrm{tr}\>\left(X\Omega Y\Omega ^\dagger\right)$ 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.

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