  This article is devoted to the asymptotic expansion of the generalized Harish
Chandra-Itzykson-Zuber matrix integral for non-unitary symmetries characterized
by 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 from
the heat kernel differential equation, for general values of k and beta. From
an expansion in terms of zonal polynomials, one obtain an expansion in powers
of the tau's for beta=1, and generalizations are considered for general beta. A
duality relation, and a transformation of products of pairs of symmetric
functions into tau polynomials, is used to obtain the expression for f(tau ij)
for general beta.
