Geometric representations of graded and rational Cherednik algebras

We provide geometric constructions of modules over the graded Cherednik algebra Hνgr and the rational Cherednik algebra Hνrat attached to a simple algebraic group G together with a pinned automorphism θ. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit C⁎-actions. In the rational Cherednik algebra case, the standard grading on these modules is derived from the perverse filtration on the cohomology of affine Springer fibers coming from its global analog: Hitchin fibers. When θ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules Lν(triv) of Hνgr and of Hνrat. We give a formula for the dimension of Lν(triv) and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details.