Geometric representations of graded and rational Cherednik algebras

Abstract

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

82 pages; 8 pictures; some minor corrections; to appear in Advances in Mathematics