The spherical part of the local and global Springer actions

TL;DR

This paper demonstrates that the spherical part of the affine Weyl group action on cohomology factors through the component group of centralizers, linking local and global Springer theories and confirming parts of existing conjectures.

Contribution

It establishes that the spherical part of the affine Weyl group action factors through the component group, connecting local and global Springer theories and verifying conjectures.

Findings

The spherical part action factors through the component group.

The result is first proved globally and then deduced locally.

Uses deformation of points on a curve to relate global and local theories.

Abstract

The affine Weyl group acts on the cohomology (with compact support) of affine Springer fibers (local Springer theory) and of parabolic Hitchin fibers (global Springer theory). In this paper, we show that in both situations, the action of the center of the group algebra of the affine Weyl group (the spherical part) factors through the action of the component group of the relevant centralizers. In the situation of affine Springer fibers, this partially verifies a conjecture of Goresky-Kottwitz-MacPherson and Bezrukavnikov-Varshavsky. We first prove this result for the global Springer action, and then deduce the result for the local Springer action from that of the global one. The argument strongly relies on the fact that we can deform points on a curve, hence giving an example of using global Springer theory to solve more classical problems.