Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quantizations of conical symplectic resolutions II: category $\mathcal
  O$ and symplectic duality   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.

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The categories of this paper are: cs
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The categories of this paper are: physics
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: atom-ph
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph