Derived equivalences for rational Cherednik algebras

Abstract

Let $W$ be a complex reflection group, and let $H_c\left(W\right)$ be the rational Cherednik algebra for $W$ depending on a parameter $c$. One can consider the category $\mathcal{O}$ for $H_c\left(W\right)$. We prove a conjecture of Rouquier that the categories $\mathcal{O}$ for $H_c\left(W\right)$ and $H_{c\text{'}}\left(W\right)$ are derived-equivalent, provided that the parameters $c,c\text{'}$ have integral difference. Two main ingredients of the proof are a connection between the Ringel duality and Harish-Chandra bimodules and an analogue of a deformation technique developed by the author and Bezrukavnikov. We also show that some of the derived equivalences we construct are perverse.