Springer theory for symplectic Galois groups

McGerty, Kevin; Nevins, Thomas

doi:10.2140/gt.2026.30.883

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Springer theory for symplectic Galois groups

Kevin McGerty and Thomas Nevins

Geometry & Topology 30 (2026) 883–928

DOI: 10.2140/gt.2026.30.883

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.

Keywords

symplectic resolutions, Weyl groups, Springer theory

Mathematical Subject Classification 2010

Primary: 14B07, 32S30

References

Publication

Received: 18 June 2019

Revised: 27 March 2024

Accepted: 4 May 2024

Published: 20 April 2026

Proposed: András I Stipsicz

Seconded: Dan Abramovich, David Fisher

Authors

Kevin McGerty
Mathematical Institute University of Oxford Oxford United Kingdom
https://www.maths.ox.ac.uk/people/kevin.mcgerty
Thomas Nevins
Department of Mathematics University of Illinois at Urbana-Champaign Urbana, IL United States

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Volume 30, issue 3 (2026)