Derived equivalences for Rational Cherednik algebras

TL;DR

This paper proves that categories O for Rational Cherednik algebras of complex reflection groups are derived equivalent when parameters differ by an integer, using duality and deformation techniques.

Contribution

It confirms Rouquier's conjecture by establishing derived equivalences between categories O for different parameters of Rational Cherednik algebras.

Findings

Categories O are derived equivalent for parameters with integral difference.

Connection established between Ringel duality and Harish-Chandra bimodules.

Some derived equivalences are shown to be perverse.

Abstract

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