Geometric representations of graded and rational Cherednik algebras

Abstract

We provide geometric constructions of modules over the graded Cherednik algebra $\mathfrak{H}^{gr}_\nu$ and the rational Cherednik algebra $\mathfrak{H}^{rat}_\nu$ attached to a simple algebraic group $\mathbb{G}$ together with a pinned automorphism $\theta$. These modules are realized on the cohomology of affine Springer fibers (of finite type) that admit $\mathbb{C}^*$-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 $\theta$ is trivial, we show that our construction gives the irreducible finite-dimensional spherical modules $\mathfrak{L}_\nu(triv)$ of $\mathfrak{H}^{gr}_\nu$ and of $\mathfrak{H}^{rat}_\nu$. We give a formula for the dimension of $\mathfrak{L}_\nu(triv)$ and give a geometric interpretation of its Frobenius algebra structure. The rank two cases are studied in further details.