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

TL;DR

This paper computes correlation functions of Harish-Chandra integrals over orthogonal and symplectic groups, transforming group integrals into Gaussian integrals over triangular matrices and deriving explicit determinantal formulas.

Contribution

It introduces a novel method to evaluate Harish-Chandra correlation functions by reducing group integrals to Gaussian integrals over triangular matrices with symmetry considerations.

Findings

Derived compact determinantal formulas for correlation functions.

Recast group integrals as Gaussian integrals over symmetric triangular matrices.

Extended the Duistermaat-Heckman theorem to these correlation functions.

Abstract

The Harish-Chandra correlation functions, i.e. integrals over compact groups of 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 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.