Cherednik algebras and Hilbert schemes in characteristic $p$

Abstract:

We prove a localization theorem for the type $\mathbf {A}_{n-1}$ rational Cherednik algebra $\mathsf {H}_c=\mathsf {H}_{1,c}(\mathbf {A}_{n-1})$ over $\overline {\mathbb {F}}_p$, an algebraic closure of the finite field. In the most interesting special case where $c\in \mathbb {F}_p$, we construct an Azumaya algebra $\scr H_c$ on $\operatorname {Hilb}^n{\mathbb {A}}^2$, the Hilbert scheme of $n$ points in the plane, such that $\Gamma (\operatorname {Hilb}^n{\mathbb {A}}^2, \scr H_c)=\mathsf {H}_c$. Our localization theorem provides an equivalence between the bounded derived categories of $\mathsf {H}_c$-modules and sheaves of coherent $\scr H_c$-modules on $\operatorname {Hilb}^n{\mathbb {A}}^2$, 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.