The Harish-Chandra correlation functions, i.e. integrals over compact groups of invariant monomials ∏ tr(Xp1 Ω Yq1 Ω† Xp2 ....) 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.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics