  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.
