Weyl group actions on the Springer sheaf

TL;DR

This paper demonstrates the equivalence of two Weyl group actions on the Springer sheaf, extending classical results to modular settings and establishing a generalized Springer correspondence.

Contribution

It generalizes the comparison of Weyl group actions on the Springer sheaf to arbitrary coefficients, including modular representations, and defines a new Springer correspondence in this context.

Findings

Weyl group actions agree up to a sign twist

Generalization to modular representation theory

Identification of zero weight spaces in small representations

Abstract

We show that two Weyl group actions on the Springer sheaf with arbitrary coefficients, one defined by Fourier transform and one by restriction, agree up to a twist by the sign character. This generalizes a familiar result from the setting of l-adic cohomology, making it applicable to modular representation theory. We use the Weyl group actions to define a Springer correspondence in this generality, and identify the zero weight spaces of small representations in terms of this Springer correspondence.