Kostka systems and exotic t-structures for reflection groups

Let W be a complex reflection group, acting on a complex vector space H. Kato has recently introduced the notion of a "Kostka system," which is a certain collection of finite-dimensional W-equivariant modules for the symmetric algebra on H. In this paper, we show that Kostka systems can be used to construct "exotic" t-structures on the derived category of finite-dimensional modules, and we prove a derived-equivalence result for these t-structures.Comment: 21 pages. v2: minor corrections; simplified proof in Section

Staggered t-structures on derived categories of equivariant coherent sheaves

Let X be a scheme, and let G be an affine group scheme acting on X. Under reasonable hypotheses on X and G, we construct a t-structure on the derived category of G-equivariant coherent sheaves that in many ways resembles the perverse coherent t-structure, but which incorporates additional information from the G-action. Under certain circumstances, this t-structure, called the "staggered t-structure," has an artinian heart, and its simple objects are particularly easy to describe. We also exhibit two small examples in which the staggered t-structure is better-behaved than the perverse coherent t-structure.Comment: 43 pages; corrected an error regarding s-structures on closed subschemes; expanded the review of equivariant derived categorie

Local Systems on Nilpotent Orbits and Weighted Dynkin Diagrams

We study the Lusztig-Vogan bijection for the case of a local system. We compute the bijection explicitly in type A for a local system and then show that the dominant weights obtained for different local systems on the same orbit are related in a manner made precise in the paper. We also give a conjecture (putatively valid for all groups) detailing how the weighted Dynkin diagram for a nilpotent orbit in the dual Lie algebra should arise under the bijection.Comment: 11 page

The affine Grassmannian and the Springer resolution in positive characteristic

An important result of Arkhipov-Bezrukavnikov-Ginzburg relates constructible sheaves on the affine Grassmannian to coherent sheaves on the dual Springer resolution. In this paper, we prove a positive-characteristic analogue of this statement, using the framework of "mixed modular sheaves" recently developed by the first author and Riche. As an application, we deduce a relationship between parity sheaves on the affine Grassmannian and Bezrukavnikov's "exotic t-structure" on the Springer resolution.Comment: 50 pages; with an appendix joint with Simon Riche. v2: minor correction

Parity sheaves on the affine Grassmannian and the Mirkovi\'c-Vilonen conjecture

We prove the Mirkovi\'c-Vilonen conjecture: the integral local intersection cohomology groups of spherical Schubert varieties on the affine Grassmannian have no p-torsion, as long as p is outside a certain small and explicitly given set of prime numbers. (Juteau has exhibited counterexamples when p is a bad prime.) The main idea is to convert this topological question into an algebraic question about perverse-coherent sheaves on the dual nilpotent cone using the Juteau-Mautner-Williamson theory of parity sheaves.Comment: 27 pages. v4: added details to Section 2 and an appendix on sheaf functors on non-locally compact space

Koszul duality and mixed Hodge modules

We prove that on a certain class of smooth complex varieties (those with "affine even stratifications"), the category of mixed Hodge modules is "almost" Koszul: it becomes Koszul after a few unwanted extensions are eliminated. We also give an equivalence between perverse sheaves on such a variety and modules for a certain graded ring, obtaining a formality result as a corollary. For flag varieties, these results were proved earlier by Beilinson-Ginzburg-Soergel using a rather different construction.Comment: 26 pages. v4: added Proposition 3.9; streamlined Section 4; other minor correction