Derived equivalence for quantum symplectic resolutions

Using techniques from the homotopy theory of derived categories and noncommutative algebraic geometry, we establish a general theory of derived microlocalization for quantum symplectic resolutions. In particular, our results yield a new proof of derived Beilinson-Bernstein localization and a derived version of the more recent microlocalization theorems of Gordon-Stafford (Gordon and Stafford in Adv Math 198(1):222-274, 2005; Duke Math J 132(1):73-135, 2006) and Kashiwara-Rouquier (Kashiwara and Rouquier in Duke Math J 144(3):525-573, 2008) as special cases. We also deduce a new derived microlocalization result linking cyclotomic rational Cherednik algebras with quantized Hilbert schemes of points on minimal resolutions of cyclic quotient singularities. © 2013 Springer Basel.