Cherednik Algebras for Algebraic Curves

Abstract

To George Lusztig with admirationFor any algebraic curve C and n≥1, Etingof introduced a “global” Cherednik algebra as a natural deformation of the cross product $\mathcal{D}({C}^{n}) \rtimes {\mathbb{S}}_{n}$ of the algebra of differential operators on C ⁿ and the symmetric group. We provide a construction of the global Cherednik algebra in terms of quantum Hamiltonian reduction. We study a category of character $\mathcal{D}$ -modules on a representation scheme associated with C and define a Hamiltonian reduction functor from that category to category $\mathcal{O}$ for the global Cherednik algebra. In the special case of the curve $C = {\mathbb{C}}^{\times }$, the global Cherednik algebra reduces to the trigonometric Cherednik algebra of type A _n−1,and our character $\mathcal{D}$-modules become holonomic $\mathcal{D}$-modules on $G{L}_{n}(\mathbb{C}) \times {\mathbb{C}}^{n}$. The corresponding perverse sheaves are reminiscent of (and include as special cases) Lusztig’s character sheaves.

Mathematics Subject Classifications (2000): 20Gxx (20C08)