Cherednik Algebras for Algebraic Curves

For any algebraic curve C and n ≥ 1, P. Etingof introduced a ‘global’ Cherednik algebra as a natural deformation of the cross product D(Cn)⋊Sn, of the algebra of differential operators on Cn 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 Dmodules on a representation scheme associated to C and define a Hamiltonian reduction functor from that category to category O for the global Cherednik algebra. In the special case of the curve C = C, the global Cherednik algebra reduces to the trigonometric Cherednik algebra of type An−1, and our character D-modules become holonomic D-modules on GLn(C) × Cn. The corresponding perverse sheaves are reminiscent of (and include as special cases) Lusztig’s character sheaves.