A generalization of Springer theory using nearby cycles

Abstract:

Let $\mathfrak g$ be a complex semisimple Lie algebra, and $f : {\mathfrak {g}} \to G \backslash \backslash {\mathfrak {g}}$ the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of $f$. 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 $\mathbb C$, for which the invariant theory works by analogy with the adjoint representations. Let $G | V$ be such a representation, $f : V \to G \backslash \backslash V$ 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 ${\mathrm {End}} (P)$ as a deformation of the group algebra ${\mathbb {C}} [W]$.