Cherednik algebras and hilbert schemesin characteristic p

We prove a localization theorem for the type An−1 rational Cherednik algebra Hc = H_{1, c}(A_n−1) over F_p, an algebraic closure of the finite field. In the most interesting special case where c ∈ F_p, we construct an Azumaya algebra Hc on Hilbn A2, the Hilbert scheme of n points in the plane, such that Γ(HilbⁿA², H_c) = H_c. Our localization theorem provides an equivalence between the bounded derived categories of H_c-modules and sheaves of coherent H_c-modules on HilbⁿA², respectively. Furthermore, we show that the Azumaya algebra splits on the formal neighborhood of each fiber of the Hilbert-Chow morphism. This provides a link between our results and those of Bridgeland, King and Reid, and Haiman.