*Algebraic Braids and the Springer Theory of the Hitchin Fibration*

Let G be a reductive group with Weyl group W. The cohomology of an (anisotropic, parabolic) Hitchin fiber with structure group G is endowed with perverse and weight filtrations. Via Springer theory, the associated bigraded vector space is a W-module. We present a new conjecture, stating that this module is controlled by a corresponding family of conjugacy classes in the Artin braid group of W. To this end, we develop a new Springer-theoretic invariant of braid conjugacy classes that, in type A, refines the HOMFLY homology of Khovanov-Rozansky. Our conjecture is therefore an identity of bigraded W-modules that generalizes an earlier conjecture of Oblomkov-Rasmussen-Shende about compactified Jacobians and HOMFLY. In certain cases, we can prove our identity by describing both sides in terms of representations of rational Cherednik algebras. If time permits, we will explain how it can be used to recover numerical P = W phenomena in the sense of nonabelian Hodge theory.