We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors.Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties. This provides a new context for known results about Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from more general resolutions.

We develop a graded version of the theory of cyclotomic q-Schur algebras, in the spirit of the work of Brundan-Kleshchev on Hecke algebras and of Ariki on q-Schur algebras. As an application, we identify the coefficients of the canonical basis on a higher level Fock space with q-analogues of the decomposition numbers of cyclotomic q-Schur algebras.We present cyclotomic q-Schur algebras as a quotient of a convolution algebra arising in the geometry of quivers -we call it quiver Schur algebra -and also diagrammatically, similar in flavor to a recent construction of Khovanov and Lauda. They are manifestly graded and so equip the cyclotomic q-Schur algebra with a non-obvious grading. On the way we construct a graded cellular basis of this algebra, resembling similar constructions for cyclotomic Hecke algebras.The quiver Schur algebra is also interesting from the perspective of higher representation theory. The sum of Grothendieck groups of certain cyclotomic quotients is known to agree with a higher level Fock space. We show that our graded version defines a higher q-Fock space (defined as a tensor product of level 1 q-deformed Fock spaces). Under this identification, the indecomposable projective modules are identified with the canonical basis and the Weyl modules with the standard basis. This allows us to prove the already described relation between decomposition numbers and canonical bases. * (X/G) is the Gequivariant Borel-Moore homology of X (for a discussion of equivariant Borel-Moore homology, see [Ful98, section 19], or [VV11, §1.2]).

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.

The degenerate Heisenberg category Heis k is a strict monoidal category which was originally introduced in the special case k = −1 by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain Z-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge −1. We prove this conjecture and extend it to arbitrary central charge k ∈ Z. We also explain how to categorify the comultiplication (generically).

We prove a Koszul duality theorem between the category of weight modules over the quantized Coulomb branch (as defined by Braverman, Finkelberg and Nakajima) attached to a group G and representation V and a category of G-equivariant D-modules on the vector space V . This is proven by relating both categories to an explicit, combinatorially presented category.These categories are related to generalized categories O for symplectic singularities. Letting O Coulomb and O Higgs be these categories for the Coulomb and Higgs branches associated to V and G, we obtain a functor O ! Coulomb → O Higgs from the Koszul dual of one to the other. This functor is an equivalence in the special cases where the hyperkähler quotient of T * V by G is a Nakajima quiver variety or smooth hypertoric variety. This includes as special cases the parabolic-singular Koszul duality of category O in type A, the categorified rank-level duality proposed by Chuang and Miyachi and proven by Shan, Vasserot and Varagnolo, and the hypertoric Koszul duality proven by Braden, Licata, Proudfoot and the author.We also show that this equivalence intertwines so-called twisting and shuffling functors. This together with the duality discussed confirms the most important components of the symplectic duality conjecture of Braden, Licata, Proudfoot and the author in this case. * (X σ,w × V X σ ′ ,w ′ ), with composition given by convolution. The Steinberg algebra is simply the sum of all the morphisms in this category; modules over the Steinberg algebra are naturally