Invariant eigendistributions on a semisimple Lie algebra and homology classes on the conormal variety. II. Representations of Weyl groups

Abstract

We construct representations of W in the homology of certain subvarieties of the cotangent bundle of the flag manifold of g, in particular in the homology of the conormal variety and of Springer varieties. Using the integral formula we prove that in top degree the former is isomorphic with the natural representation of W on coherent families of invariant eigendistributions and decomposes into a sum of the latter according to the decomposition of the nilpotent variety into G 0-orbits. As consequences we obtain Springer's theorem on irreducible representations of Weyl groups, a formula for irreducible characters of G 0 as integrals over characteristic varieties of D modules, an identification of the harmonic polynomials occuring in the asymptotic expansion at zero of invariant eigendistributions as homology classes on the flag manifold, a formula for the Fourier transforms of nilpotent G 0 orbits, and a proof of a conjecture of Joseph on the characteristic polynomials of intersections of nilpotent G 0-orbits with n 0.