Modular generalized Springer correspondence III: exceptional groups

TL;DR

This paper completes the modular generalized Springer correspondence for all connected reductive groups, providing a uniform proof, algorithms, and detailed classifications for exceptional groups, especially in good characteristic.

Contribution

It offers a uniform proof for the disjointness of induction series and classifies cuspidal pairs for exceptional groups in good characteristic.

Findings

Disjointness of induction series established uniformly

Algorithm for minimal cuspidal data determination

Complete classification of cuspidal pairs in exceptional groups

Abstract

We complete the construction of the modular generalized Springer correspondence for an arbitrary connected reductive group, with a uniform proof of the disjointness of induction series that avoids the case-by-case arguments for classical groups used in previous papers in the series. We show that the induction series containing the trivial local system on the regular nilpotent orbit is determined by the Sylow subgroups of the Weyl group. Under some assumptions, we give an algorithm for determining the induction series associated to the minimal cuspidal datum with a given central character. We also provide tables and other information on the modular generalized Springer correspondence for quasi-simple groups of exceptional type, including a complete classification of cuspidal pairs in the case of good characteristic, and a full determination of the correspondence in type $G_2$.