A Generalization of Springer Theory Using Nearby Cycles

Let g be a complex semisimple Lie algebra, and f : g → G\\g the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of f . In this paper, we give a generalization of Springer theory to visible, polar representations. It is a class of rational representations of reductive groups over C, for which the invariant theory works by analogy with the adjoint representations. Let G |V be such a representation, f : V → G\\V the quotient map, and P the sheaf of nearby cycles of f . We show that the Fourier transform of P is an intersection homology sheaf on V ∗. Associated to G |V , there is a finite complex reflection group W , called the Weyl group of G |V . We describe the endomorphism ring End(P) as a deformation of the group algebra C[W ].