An extension of the HarishChandra-Itzykson-Zuber integral

The HarishChandra-Itzykson-Zuber integral over the unitary group U(k) (beta = 2) is present in numerous problems involving Hermitian random matrices. It is well known that the result is semi-classically exact. This simple result does not extend to other symmetry groups, such as the symplectic or orthogonal groups. In this article the analysis of this integral is extended first to the symplectic group Sp(k) (beta=4). There the semi-classical approximation has to be corrected by a WKB expansion. It turns out that this expansion stops after a finite number of terms; in other words the WKB approximation is corrected by a polynomial in the appropriate variables. The analysis is based upon new solutions to the heat kernel differential equation. We have also investigated arbitrary values of the parameter beta ,which characterizes the symmetry group. Closed formulae are derived for arbitrary beta and k = 3, and also for large beta and arbitrary k.