WKB-Expansion of the HarishChandra-Itzykson-Zuber Integral for Arbitrary

This article is devoted to the asymptotic expansion of the generalized HarishChandra-Itzykson-Zuber matrix integral for non-unitary symmetries characterizedby a parameter beta(as usual beta =1,2 and 4 correspond to the orthogonal,unitary and symplectic group integrals). A WKB-expansion for f is derived fromthe heat kernel differential equation, for general values of k and beta. Froman expansion in terms of zonal polynomials, one obtain an expansion in powersof the tau's for beta=1, and generalizations are considered for general beta. Aduality relation, and a transformation of products of pairs of symmetricfunctions into tau polynomials, is used to obtain the expression for f(tau ij)for general beta.Comment: 72 page