Parity sheaves and tilting modules

Juteau, Daniel; Mautner, Carl; Williamson, Geordie

doi:10.24033/asens.2282

Cited by 20 publications

(35 citation statements)

References 18 publications

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“…Their proof requires the characteristic of k to be very good for G ∨ , but a priori not necessarily a JMW prime. In fact, their work implies that every good prime is a JMW prime, improving on the bounds established [JMW2,Theorem 1.8]. As a consequence, the main result of [AR] and Theorem 1.1 of the present paper both hold in good characteristic.…”

Section: 4supporting

confidence: 62%

“…The main result of the paper is the following modular analogue of the result of [ABG]. Recall that a JMW prime for G ∨ is a good prime such that the main result of [JMW2] holds in that characteristic: that is, S sends tilting G ∨ -modules to spherical parity sheaves. (Recently, Mautner and Riche have shown that every good prime is a JMW prime; see §1.4 below).…”

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confidence: 95%

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The affine Grassmannian and the Springer resolution in positive characteristic

Achar¹,

Rider²

2016

Compositio Math.

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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...

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Section: 4supporting

confidence: 62%

mentioning

confidence: 95%

The affine Grassmannian and the Springer resolution in positive characteristic

Achar¹,

Rider²

2016

Compositio Math.

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“…It also follows from the results of [JMW2] that the equivalence S restricts to an equivalence of categories…”

Section: Now We Setmentioning

confidence: 94%

“…Moreover, any indecomposable parity complex in ParityǦ (O) (Gr, F) is isomorphic to E λ [i] for some unique λ ∈ X + and i ∈ Z. By [MR2, Corollary 1.6], the parity complexes E λ are perverse (see also [JMW2] for an earlier proof of this fact, under stronger assumptions). We denote by PParityǦ (O) (Gr, F) the full subcategory of ParityǦ (O) (Gr, F) consisting of parity complexes which are perverse, in other words of direct sums of objects E λ (with no shift).…”

Section: Now We Setmentioning

confidence: 98%

“…If L is the Levi subgroup of G whose roots are α and −α, then we have a Satake equivalence SĽ forĽ, with dual group L, and RǦ L corresponds, under S and SĽ, to the restriction functor Rep(G) → Rep(L). It is also known that RǦ L sends parity complexes to parity complexes, see [JMW2,Theorem 1.6]. Using isomorphism (5.2.2) this provides a canonical isomorphism…”

Section: Composing With the Natural Morphism Hmentioning

confidence: 99%

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