1. Introduction

The remarkable and useful formula has been known for the last two decades; see [Itzykson and Zuber 1980; Mehta 1981; Mehta 1991, Appendix A.5]. Here A and A’ are nxn complex hermitian matrices having eigenvalues x := ﹛x1,…,xn﹜ and respectively, integration is over the n×n complex unitary matrices U with the invariant Haar measure dU normalized such that. The function △(x) is the product of differences of the Xj:

We would like to have a similar formula when A and A’ are n×n real symmetric or quaternion self-dual matrices and the integration is over n×n real orthogonal or quaternion symplectic matrices U, a formula not presently known. These three cases are usually denoted by a parameter (3 taking values 1, 2 and 4 corresponding respectively to the integration over n x n real orthogonal, complex unitary and quaternion symplectic matrices U. We will show that the integral in (1-1) with a measure dU invariant under the appropriate group can be expressed in terms of the eigenvalues and eigenfunctions of a particular hamiltonian. This hamiltonian is closely related to the Calogero model [1969a; 1969b; 1971] where one considers the quantum n-body problem with the hamiltonian.