Localization theorems for quantized symplectic resolutions

TL;DR

This paper extends localization theorems to quantized symplectic resolutions, specifically proving full localization for certain Nakajima quiver varieties, enhancing understanding of their categorical structures.

Contribution

It establishes Beilinson-Bernstein type localization theorems for quantizations of conical symplectic resolutions, with new results for Nakajima quiver varieties.

Findings

Proves full localization theorems for finite and affine type A Nakajima quiver varieties.

Provides conditions for the global section functor to be an equivalence on category .

Introduces criteria involving tilting generators for exactness of global sections.

Abstract

The goal of this paper is to establish Beilinson-Bernstein type localization theorems for quantizations of some conical symplectic resolutions. We prove the full localization theorems for finite and affine type A Nakajima quiver varieties. The proof is based on two partial results that hold in more general situations. First, we establish an exactness result for global section functor if there is a tilting generator that has a rank 1 summand. Second, we examine when the global section functor restricts to an equivalence between categories $\mathcal{O}$.