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

Abstract

The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials $\prod \operatorname{tr}(X^{p_{1}}\varOmega Y^{q_{1}}\varOmega^{\dagger}X^{p_{2}}\cdots)$ 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.