  Let G_R be a Lie group acting on an oriented manifold M, and let $\omega$ be
an equivariantly closed form on M. If both G_R and M are compact, then the
integral $\int_M \omega$ is given by the fixed point integral localization
formula (Theorem 7.11 in [BGV]). Unfortunately, this formula fails when the
acting Lie group G_R is not compact: there simply may not be enough fixed
points present. A proposed remedy is to modify the action of G_R in such a way
that all fixed points are accounted for.
  Let G_R be a real semisimple Lie group, possibly noncompact. One of the most
important examples of equivariantly closed forms is the symplectic volume form
$d\beta$ of a coadjoint orbit $\Omega$. Even if $\Omega$ is not compact, the
integral $\int_{\Omega} d\beta$ exists as a distribution on the Lie algebra
g_R. This distribution is called the Fourier transform of the coadjoint orbit.
  In this article we will apply the localization results described in [L1] and
[L2] to get a geometric derivation of Harish-Chandra's formula (9) for the
Fourier transforms of regular semisimple coadjoint orbits. Then we will make an
explicit computation for the coadjoint orbits of elements of G_R* which are
dual to regular semisimple elements lying in a maximally split Cartan
subalgebra of g_R.
