Wall-crossing functors for quantized symplectic resolutions: perversity and partial Ringel dualities

TL;DR

This paper investigates wall-crossing functors for quantized symplectic resolutions, establishing their perverse equivalence properties, verifying a conjecture for Nakajima quiver varieties, and relating them to Ringel duality in categories O.

Contribution

It proves wall-crossing functors are perverse equivalences, verifies a conjecture for Nakajima quiver varieties, and introduces new stratified structures linking wall-crossing to Ringel duality.

Findings

Wall-crossing functors are perverse equivalences.

Verification of an Etingof type conjecture for Nakajima quiver varieties.

Introduction of standardly stratified structures and their relation to Ringel duality.

Abstract

In this paper we study wall-crossing functors between categories of modules over quantizations of symplectic resolutions. We prove that wall-crossing functors through faces are perverse equivalences and use this to verify an Etingof type conjecture for quantizations of Nakajima quiver varieties associated to affine quivers. In the case when there is a Hamiltonian torus action on the resolution with finitely many fixed points so that it makes sense to speak about categories $\mathcal{O}$ over quantizations, we introduce new standardly stratified structures on these categories $\mathcal{O}$ and relate the wall-crossing functors to the Ringel duality functors associated to these standardly stratified structures.