Abstract. 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.1. Introduction 1.1. Main result. Let G be a connected reductive complex algebraic group, and let G ∨ be the Langlands dual group over an algebraically closed field k. Recall that the geometric Satake equivalence is an equivalence of tensor abelian categorieswhere Rep(G ∨ ) is the category of finite-dimensional rational representations of G ∨ , and Perv GO (Gr, k) is the category of spherical perverse k-sheaves on the affine Grassmannian Gr. When k = C, there is an extensive body of work (see [AB,ABG,B3,BF], among others) exhibiting various ways of extending S to an equivalence of derived or triangulated categories. In particular, an important theorem due to Arkhipov-Bezrukavnikov-Ginzburg [ABG] relates Iwahori-constructible sheaves on Gr to coherent sheaves on the Springer resolutionÑ for G ∨ . In this paper, we begin the project of studying derived versions of (1.1) in positive characteristic. We work in the framework of "mixed modular derived categories" recently developed by the first author and S. Riche [ARc2, ARc3]. The main result of the paper is the following modular analogue of the result of [ABG]. Moreover, this equivalence is compatible with the geometric Satake equivalence: for any Readers who are familiar with [ABG] will recognize a number of familiar ingredients in this paper, including Wakimoto sheaves; the ind-perverse sheaf corresponding to the regular representation; and realizations of the coordinate rings of N andÑ as Ext-algebras on Gr. However, the behavior of these objects is often more complicated than in [ABG], both because of the nonsemisimplicity of the representation theory of G ∨ , and because mixed modular sheaves are harder to work with than mixedQ ℓ -sheaves.One salient difficulty with mixed modular sheaves is that it is not known whether there is a well-behaved "forgetful" functor D, so we cannot compare mixed and ordinary perverse k-sheaves. As a consequence, a key construction of [ABG], giving a dg-model for D mix (I) (Gr,Q ℓ ) in terms of projective pro-perverse sheaves, cannot be carried out in positive characteristic. Instead, we use a dg-model for D mix (I) (Gr, k) based on parity sheaves. (Indeed, perverse sheaves are almost absent from this paper.) The lack of a forgetful functor also means that unlike in [ABG], we do not know how to deduce a non-mixed analogue of Theorem 1.1, describing the ordinary derived category D b (I) (Gr, k). In [ABG], the result we have been discussing is used as a step in the proof that Perv (I) (Gr, C) is equivalent to the principal block o...
We prove the Mirković-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.
Recall that the Springer correspondence relates representations of the Weyl group to perverse sheaves on the nilpotent cone. We explain how to extend this to an equivalence between the triangulated category generated by the Springer perverse sheaf and the derived category of differential graded modules over a dg-ring related to the Weyl group.
In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q ℓ , or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.
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