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

Let GR be a Lie group acting on an oriented manifold M, and let ω be an equivariantly closed form on M. If both GR and M are compact, then the integral ∫Mω is given by the fixed point integral localization formula (Theorem 7.11 in Berline et al. Heat Kernels and Dirac Operators, Springer, Berlin, 1992). Unfortunately, this formula fails when the acting Lie group GR is not compact: there simply may not be enough fixed points present. A proposed remedy is to modify the action of GR in such a way that all fixed points are accounted for. Let GR be a real semisimple Lie group, possibly noncompact. One of the most important examples of equivariantly closed forms is the symplectic volume form dβ of a coadjoint orbit Ω. Even if Ω is not compact, the integral ∫Ωdβ exists as a distribution on the Lie algebra gR. This distribution is called the Fourier transform of the coadjoint orbit. In this article, we will apply the localization results described in [L1,L2] to get a geometric derivation of Harish-Chandraʹs 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 gR∗ which are dual to regular semisimple elements lying in a maximally split Cartan subalgebra of gR.