A fixed point localization formula for the Fourier transform of regular semisimple coadjoint orbits

Abstract

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.