Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirkovic-Vilonen conjecture

Abstract

Let $\mathbf{G}$ be a connected reductive group over an algebraically closed field $\mathbb{F}$ of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of $\mathbf{G}$, and Iwahori-constructible $\mathbb{F}$-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirkovic-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence.

v1: 66 pages; v2: 67 pages, minor corrections; v3: 65 pages, final version, to appear in JEMS

References in corpus (3)

• Modular representations of reductive groups and geometry of affine Grassmannians
• Differential operators on G/U and the affine Grassmannian
• Parity sheaves on the affine Grassmannian and the Mirković-Vilonen conjecture

Cited by in corpus (6)

• Mixed modular perverse sheaves on moment graphs
• Mixed perverse sheaves on flag varieties of Coxeter groups
• Kostant section, universal centralizer, and a modular derived Satake equivalence
• Notes on exotic and perverse-coherent sheaves
• Filtrations of tilting modules and costalks of parity sheaves
• Exotic t-structure for partial resolutions of the nilpotent cone