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

TL;DR

The paper develops a broad representation-theoretic framework for conical symplectic resolutions by defining and studying category $\mathcal{O}$ in this geometric setting. It establishes intrinsic structures such as highest-weight and (conjectural) Koszul properties for the algebraic and geometric categories $\mathcal{O}_{\!a}$ and $\mathcal{O}_{\!g}$, and introduces twisting and shuffling functors that generate braid-group actions and braid group symmetries on these categories. A central theme is symplectic duality, a Koszul-duality-based correspondence between pairs of resolutions that exchanges twisting and shuffling, with rich cohomological and geometric consequences including identifications of weight spaces via Nakajima/Ginzburg–MV–type constructions and potential connections to mirror-symmetry-like physics. The paper provides extensive examples (cotangent bundles, hypertoric varieties, S3-varieties, quiver varieties, and affine Grassmannian slices) and develops filtrations, cell decompositions, and characteristic-cycle technology that unify and extend known stories in geometric representation theory. Losev’s appendix anchors the highest-weight property in one key case, supporting the broad program of linking geometric and representation-theoretic structures through symplectic duality and its cohomological ramifications.

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/Mirković-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.

Theorems & Definitions (206)

• Remark 2.1
• Remark 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Remark 2.6
• Remark 2.7
• Theorem 2.8
• Theorem 2.9
• Definition 2.10