Quantizations of conical symplectic resolutions II: category $\mathcal O$ and symplectic duality
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/MirkovićVilonen, of weight spaces of simple representations of simplylaced simple algebraic groups. An appendix by Ivan Losev establishes a key step in the proof that $\mathcal O$ is highest weight.
 Publication:

arXiv eprints
 Pub Date:
 July 2014
 arXiv:
 arXiv:1407.0964
 Bibcode:
 2014arXiv1407.0964B
 Keywords:

 Mathematics  Representation Theory;
 Mathematics  Algebraic Geometry;
 Mathematics  Symplectic Geometry
 EPrint:
 118 pages. v3: mostly changes based on new material which has appeared since last version