# Springer theory for symplectic Galois groups

## TL;DR

This paper extends Springer's classical theory to symplectic resolutions of singularities, analyzing Weyl group actions on cohomology in various geometric contexts.

## Contribution

It generalizes Springer's theory to broader symplectic resolutions and explores Weyl group actions in affine quiver varieties and related dual examples.

## Key findings

- Constructed Weyl group actions on cohomology of $ADE$ quiver varieties.
- Analyzed symplectic geometry features of quiver varieties.
- Documented properties of symplectic geometry in these contexts.

## Abstract

A classical and beautiful story in geometric representation theory is the construction by Springer of an action of the Weyl group on the cohomology of the fibres of the Springer resolution of the nilpotent cone. We establish a natural extension of Springer's theory to arbitrary symplectic resolutions of conical symplectic singularities. We analyse features of the action in the case of affine quiver varieties, constructing Weyl group actions on the cohomology of $ADE$ quiver varieties, and also consider "symplectically dual" examples arising from slices in the affine Grassmannian. Along the way, we document some basic features of the symplectic geometry of quiver varieties.