Classical Springer representations are given by the action of the Weyl group on the cohomology of certain subvarieties of the flag variety called Springer fibers. In a series of two talks, the first of which will be expository, we will construct a "global" analogue of this action. We view the (properly modified) Hitchin moduli space as the global analogue of the Grothendieck-Springer resolution.
The rich symmetry of the Hitchin fibration enables us to construct an action of the (graded) double affine Hecke algebra (DAHA) on the cohomology of parabolic Hitchin fibers. This construction was motivated by Ngo's proof of the fundamental lemma. It has applications to the harmonic analysis of p-adic groups and the representation theory of the DAHAs, and also has connections with geometric Langlands duality.