Quantizations of conical symplectic resolutions II: category $\mathcal O$ and symplectic duality

TL;DR

This paper generalizes the classical category O to symplectic resolutions, explores its properties, and introduces symplectic duality, revealing deep connections between representation theory and geometry.

Contribution

It defines and studies a new category O for symplectic resolutions, establishing its properties and introducing the concept of symplectic duality with geometric and algebraic implications.

Findings

Category O often exhibits Koszulity.

Koszul duality relates categories O of dual resolutions.

Symplectic duality links geometric and representation-theoretic structures.

Abstract

We define and study category $\mathcal O$ for a symplectic resolution, generalizing the classical BGG category $\mathcal O$, which is associated with the Springer resolution. This includes the development of intrinsic properties parallelling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of individual examples. We observe that category $\mathcal O$ is often Koszul, and its Koszul dual is often equivalent to category $\mathcal O$ for a different symplectic resolution. This leads us to define the notion of a symplectic duality between symplectic resolutions, which is a collection of isomorphisms between representation theoretic and geometric structures, including a Koszul duality between the two categories. This duality has various cohomological consequences, including (conjecturally) an identification of two geometric realizations, due to Nakajima and Ginzburg/Mirkovi\'c-Vilonen, of weight spaces of simple representations of simply-laced simple algebraic groups. An appendix by Ivan Losev establishes a key step in the proof that $\mathcal O$ is highest weight.