A generalization of Springer theory using nearby cycles

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 ${\mathbb{C}} [W]$.