Quantizations of conical symplectic resolutions II: category O and symplectic duality

We define and study category O for a symplectic resolution, generalizing the classical BGG category O, which is associated with the Springer resolution. This includes the development of intrinsic properties paralleling 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 O is often Koszul, and its Koszul dual is often equivalent to category 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 O is highest weight. Supported by NSA grants H98230-08-1-0097 and H98230-11-1-0180. Supported by an ARC Discovery Early Career fellowship. Supported by NSF grant DMS-0950383. Supported by NSF grant DMS-1151473.