Let be a complex reflection group, and let be the rational Cherednik algebra for depending on a parameter . One can consider the category for . We prove a conjecture of Rouquier that the categories for and are derived-equivalent, provided that the parameters 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.
"Derived equivalences for rational Cherednik algebras." Duke Math. J. 166 (1) 27 - 73, 15 January 2017. https://doi.org/10.1215/00127094-3674223