Generalized Springer Theory for D-modules on a Reductive Lie Algebra

TL;DR

This paper extends the Springer theory to a broader setting of D-modules on a reductive Lie algebra, providing a categorical decomposition related to cuspidal data and developing new functorial tools.

Contribution

It introduces a generalized framework for Springer theory for D-modules on Lie algebras, including a block decomposition and functorial constructions.

Findings

Categorical decomposition into blocks indexed by cuspidal data.

Equivalence of blocks to D-modules on the center with Weyl group action.

Development of parabolic induction and restriction functors.

Abstract

Given a reductive group $G$, we give a description of the abelian category of $G$-equivariant $D$-modules on $\mathfrak{g}=\mathrm{Lie}(G)$, which specializes to Lusztig's generalized Springer correspondence upon restriction to the nilpotent cone. More precisely, the category has an orthogonal decomposition in to blocks indexed by cuspidal data $(L,\mathcal{E})$, consisting of a Levi subgroup $L$, and a cuspidal local system $\mathcal{E}$ on a nilpotent $L$-orbit. Each block is equivalent to the category of $D$-modules on the center $\mathfrak{z}(\mathfrak{l})$ of $\mathfrak{l}$ which are equivariant for the action of the relative Weyl group $N_G(L)/L$. The proof involves developing a theory of parabolic induction and restriction functors, and studying the corresponding monads acting on categories of cuspidal objects. It is hoped that the same techniques will be fruitful in understanding similar questions in the group, elliptic, mirabolic, quantum, and modular settings.