Geometric representations of graded and rational Cherednik algebras

Abstract

We provide geometric constructions of modules over the graded Cherednik algebra ${\mathfrak{H}}_{\nu }^{gr}$ and the rational Cherednik algebra ${\mathfrak{H}}_{\nu }^{rat}$ 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 ${\mathbb{C}}^{\ast }$-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 ${\mathfrak{H}}_{\nu }^{gr}$ and of ${\mathfrak{H}}_{\nu }^{rat}$. 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.

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

References in corpus (6)

• Rouquier's conjecture and diagrammatic algebra
• Finite dimensional representations of rational Cherednik algebras
• Proof of Varagnolo-Vasserot conjecture on cyclotomic categories O
• Heisenberg algebras and rational double affine Hecke algebras
• The spherical part of the local and global Springer actions
• Fibration de Hitchin et endoscopie

Cited by in corpus (2)

• Character formulas and Bernstein-Gelfand-Gelfand resolutions for Cherednik algebra modules
• A combinatorial formula for the nabla operator