Global Springer theory

We generalize Springer representations to the context of groups over a global function field. The global counterpart of the Grothendieck simultaneous resolution is the parabolic Hitchin fibration. We construct an action of the graded double affine Hecke algebra (DAHA) on the direct image complex of the parabolic Hitchin fibration. In particular, we get representations of the degenerate graded DAHA on the cohomology of parabolic Hitchin fibers, providing the first step towards a global Springer theory.