  Recently the correlation functions of the so-called
Itzykson-Zuber/Harish-Chandra integrals were computed (by one of the authors
and collaborators) for all classical groups using an integration formula that
relates integrals over compact groups with respect to the Haar measure and
Gaussian integrals over a maximal nilpotent Lie subalgebra of their
complexification. Since the integration formula a posteriori had the same form
for the classical series, a conjecture was formulated that such a formula
should hold for arbitrary semisimple Lie groups. We prove this conjecture using
an abstract Lie-theoretic approach.
