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 and Kashiwara-Rouquier 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.