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.