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.

Abstract. We associate to every matroid M a polynomial with integer coefficients, which we call the Kazhdan-Lusztig polynomial of M , in analogy with Kazhdan-Lusztig polynomials in representation theory. We conjecture that the coefficients are always non-negative, and we prove this conjecture for representable matroids by interpreting our polynomials as intersection cohomology Poincaré polynomials. We also introduce a q-deformation of the Möbius algebra of M , and use our polynomials to define a special basis for this deformation, analogous to the canonical basis of the Hecke algebra. We conjecture that the structure coefficients for multiplication in this special basis are non-negative, and we verify this conjecture in numerous examples.

Given a hyperplane arrangement in an affine space equipped with a linear functional, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O. LINEAR PROGRAMMING Polarized arrangements.Let I be a finite set. Definition 2.1. A polarized arrangement indexed by I is a triple V = (V, η, ξ) consisting of • a vector subspace V ⊂ R I , • a vector η ∈ R I /V , and • a covector ξ ∈ V * = (R I ) * /V ⊥ , such that (a) every lift of η to R I has at least |I| − dim V non-zero entries, and (b) every lift of ξ to (R I ) * has at least dim V non-zero entries.

We consider an orbifold X obtained by a Kähler reduction of C n , and we define its "hyperkähler analogue" M as a hyperkähler reduction of T * C n ∼ = H n by the same group. In the case where the group is abelian and X is a toric variety, M is a toric hyperkähler orbifold, as defined in [BD], and further studied in [K1, K2] and [HS]. The variety M carries a natural action of S 1 , induced by the scalar action of S 1 on the fibers of T * C n . In this paper we study this action, computing its fixed points and its equivariant cohomology. As an application, we use the associated Z 2 action on the real locus of M to compute a deformation of the Orlik-Solomon algebra of a smooth, real hyperplane arrangement H, depending nontrivially on the affine structure of the arrangement. This deformation is given by the Z 2 -equivariant cohomology of the complement of the complexification of H, where Z 2 acts by complex conjugation.

Abstract. Hypertoric varieties are quaternionic analogues of toric varieties, important for their interaction with the combinatorics of matroids as well as for their prominent place in the rapidly expanding field of algebraic symplectic and hyperkähler geometry. The aim of this survey is to give clear definitions and statements of known results, serving both as a reference and as a point of entry to this beautiful subject.Given a linear representation of a reductive complex algebraic group G, there are two natural quotient constructions. First, one can take a geometric invariant theory (GIT) quotient, which may also be interpreted as a Kähler quotient by a maximal compact subgroup of G. Examples of this sort include toric varieties (when G is abelian), moduli spaces of spacial polygons, and, more generally, moduli spaces of semistable representations of quivers. A second construction involves taking an algebraic symplectic quotient of the cotangent bundle of V , which may also be interpreted as a hyperkähler quotient. The analogous examples of the second type are hypertoric varieties, hyperpolygon spaces, and Nakajima quiver varieties.The subject of this survey will be hypertoric varieties, which are by definition the varieties obtained from the second construction when G is abelian. Just as the geometry and topology of toric varieties is deeply connected to the combinatorics of polytopes, hypertoric varieties interact richly with the combinatorics of hyperplane arrangements and matroids. Furthermore, just as in the toric case, the flow of information goes in both directions.On one hand, Betti numbers of hypertoric varieties have a combinatorial interpretation, and the geometry of the varieties can be used to prove combinatorial results. Many purely algebraic constructions involving matroids acquire geometric meaning via hypertoric varieties, and this has led to geometric proofs of special cases of the g-theorem for matroids [HSt, 7.4] and the Kook-Reiner-Stanton convolution formula [PW, 5.4]. Future plans include a geometric interpretation of the Tutte polynomial and of the phenomenon of Gale duality of matroids [BLP].On the other hand, hypertoric varieties are important to geometers with no interest in combinatorics simply because they are among the most explicitly understood examples of algebraic symplectic or hyperkähler varieties, which are becoming increasingly prevalent in many areas of mathematics. For example, Nakajima's quiver varieties include resolutions of Slodowy slices and Hilbert schemes of points on ALE spaces, both of which play major roles in modern representation theory. Moduli spaces of Higgs bundles are currently receiving a lot of attention in string theory, and character varieties of fundamental groups of surfaces and 3-manifolds have become an important tool in low-dimensional topology. Hypertoric