On categories O for quantized symplectic resolutions

TL;DR

This paper investigates categories O associated with quantized symplectic resolutions with Hamiltonian torus actions, establishing stratified structures and showing that certain functors act as derived equivalences linked to hyperplane arrangements.

Contribution

It introduces standardly stratified structures on categories O for quantized symplectic resolutions and proves that cross-walling functors are derived equivalences forming an action of the Deligne groupoid.

Findings

Standardly stratified structures are established on categories O.

Cross-walling functors are proven to be derived equivalences.

These functors form an action of the Deligne groupoid.

Abstract

In this paper we study categories O over quantizations of symplectic resolutions admitting Hamiltonian tori actions with finitely many fixed points. In this generality, these categories were introduced by Braden, Licata, Proudfoot and Webster. We establish a family of standardly stratified structures (in the sense of the author and Webster) on these categories O. We use these structures to study shuffling functors of Braden, Licata, Proudfoot and Webster (called cross-walling functors in this paper). Most importantly, we prove that all cross-walling functors are derived equivalences that define an action of the Deligne groupoid of a suitable real hyperplane arrangement.