Constructible sheaves on nilpotent cones in rather good characteristic

TL;DR

This paper explores modular Springer theory for complex reductive groups in good characteristic, establishing comparison theorems, verifying conjectures, and classifying sheaves in the nilpotent cone.

Contribution

It provides a comparison between characteristic- and characteristic-0 Springer correspondences and advances understanding of sheaves in modular settings.

Findings

Comparison theorem relating modular and classical Springer correspondence

Verification of Mautner's cleanness conjecture in some cases

Classification of supercuspidal sheaves and derived category decomposition

Abstract

We study some aspects of modular generalized Springer theory for a complex reductive group $G$ with coefficients in a field $\mathbb k$ under the assumption that the characteristic $\ell$ of $\mathbb k$ is rather good for $G$, i.e., $\ell$ is good and does not divide the order of the component group of the centre of $G$. We prove a comparison theorem relating the characteristic-$\ell$ generalized Springer correspondence to the characteristic-$0$ version. We also consider Mautner's characteristic-$\ell$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.