Derived equivalence for quantum symplectic resolutions

TL;DR

This paper develops a comprehensive theory of derived microlocalization for quantum symplectic resolutions, unifying and extending existing localization theorems and linking algebraic structures with geometric resolutions.

Contribution

It introduces a general framework for derived microlocalization, providing new proofs of classical localization theorems and establishing novel connections between Cherednik algebras and quantized Hilbert schemes.

Findings

Unified derived microlocalization theory for quantum symplectic resolutions

New proof of derived Beilinson-Bernstein localization

Link between cyclotomic Cherednik algebras and quantized Hilbert schemes

Abstract

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.