  We reconsider the problem of calculating a general spectral correlation
function containing an arbitrary number of products and ratios of
characteristic polynomials for a N x N random matrix taken from the Gaussian
Unitary Ensemble (GUE).
  Deviating from the standard "supersymmetry" approach, we integrate out
Grassmann variables at the early stage and circumvent the use of the
Hubbard-Stratonovich transformation in the "bosonic" sector. The method,
suggested recently by one of us, is shown to be capable of calculation when
reinforced with a generalization of the Itzykson-Zuber integral to a
non-compact integration manifold. We arrive to such a generalisation by
discussing the Duistermaat-Heckman localization principle for integrals over
non-compact homogeneous Kaehler manifolds.
  In the limit of large $N$ the asymptotic expression for the correlation
function reproduces the result outlined earlier by Andreev and Simons.
