A topological approach to induction theorems in Springer theory

TL;DR

This paper presents a topological framework for understanding induction theorems in Springer theory, extending previous results to broader contexts including mod p cohomology and general centralizers.

Contribution

It introduces a topological construction that lifts Springer group actions to homotopy types and generalizes induction theorems to new settings.

Findings

Lifted Springer actions to homotopy types

Generalized induction theorems for broader cases

Extended Springer representations to mod p cohomology

Abstract

We give a self-contained account of a construction due to Rossmann which lifts Springer's action of a Weyl group on the cohomology of a Springer fiber to an action on its homotopy type. We use this construction to produce a generalization of an "induction theorem" of Alvis and Lusztig, which relates the Springer representations attached to a reductive group to those attached to a Levi subgroup. Our generalization applies to more general centralizers and to representations of Weyl groups on mod p cohomology.