A generalization of Springer theory using nearby cycles

Abstract

Let g \mathfrak g be a complex semisimple Lie algebra, and f : g → G ∖ ∖ g f : {\mathfrak {g}} \to G \backslash \backslash {\mathfrak {g}} the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of f 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 \mathbb C , for which the invariant theory works by analogy with the adjoint representations. Let G | V G \, | \, V be such a representation, f : V → G ∖ ∖ V f : V \to G \backslash \backslash V the quotient map, and P P the sheaf of nearby cycles of f f . We show that the Fourier transform of P P is an intersection homology sheaf on V ∗ V^* . Associated to G | V G \, | \, V , there is a finite complex reflection group W W , called the Weyl group of G | V G \, | \, V . We describe the endomorphism ring E n d ( P ) {\mathrm {End}} (P) as a deformation of the group algebra C [ W ] {\mathbb {C}} [W] .