Parity sheaves on the affine Grassmannian and the Mirković–Vilonen conjecture

Achar, Pramod N.; Rider, Laura

doi:10.1007/s11511-016-0132-6

Cited by 23 publications

(28 citation statements)

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“…Hence our claim follows from the analogous claim in the case of F-vector spaces, which is standard. 4 In [12], Bernstein and Lunts work in a topological setting. However their constructions can be adapted to the algebraic setting (in particular to the setting ofétale derived categories), see e.g.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Modular perverse sheaves on flag varieties I: tilting and parity sheaves

Achar¹,

Simón²

2016

Ann. Sci. École Norm. Sup.

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Abstract. In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group G is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group G, for any field of coefficients whose characteristic is good for G. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.1. Introduction 1.1. This paper is the first in a series devoted to investigating the structure of the category of Bruhat-constructible perverse sheaves on the flag variety of a complex connected reductive algebraic group, with coefficients in a field of positive characteristic. In this part, adapting some constructions of Bezrukavnikov-Yun [13] in the characteristic 0 setting, we show that in good characteristic, the category of parity sheaves on the flag variety of a reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the Langlands dual group. We also derive a number of interesting consequences of this result, in particular on the computation of multiplicities of simple perverse sheaves in standard perverse sheaves, on Soergel's "modular category O," and on decomposition numbers.1.2. Some notation. Let G be a complex connected reductive algebraic group, and let T ⊂ B ⊂ G be a maximal torus and a Borel subgroup. The choice of B determines a choice of positive roots of (G, T ), namely those appearing in Lie(B). Consider also the Langlands dual dataŤ ⊂B ⊂Ǧ. That is,Ǧ is a complex connected reductive group, and we are given an isomorphism X * (T ) ∼ = X * (Ť ) which identifies the roots of G with the coroots ofǦ (and the positive roots determined by B with the positive coroots determined byB).We are interested in the varieties B := G/B andB :=Ǧ/B, in the derived categories D of sheaves of k-vector spaces on these varieties, constructible with respect to the stratification by B-orbits, resp.B-orbits (where k is field), and in their abelian subcategories P (B) (B, k), resp. P (B) (B, k) of perverse sheaves (for the middle perversity). The category P (B) (B, k) is highest weight, with simple objects {IC w , w ∈ W }, standard objects {∆ w , w ∈ W }, costandard objects {∇ w , w ∈ W }, indecomposable projective objects {P w , w ∈ W } and indecomposable tilting objects {T w , w ∈ W } naturally parametrized by the Weyl group W of (G, T ). Similar remarks apply of course to P (B) (B, k), and we denote the corresponding objects byǏC w ,∆ w ,∇ w ,P w ,Ť w . (Note that the Weyl group of (Ǧ,Ť ) is canonically identified with W .)1.3. The case k = C. These categories have been extensively studied in the case k = C: see in particular [10,9,13]. To state some of their properties we need some notation. We will denote by IC (B) (B, C) the additive category of semisimple objects in D (B, C) (i.e. the full subcategory whose objects are direct sums of shifts of simple perverse sheaves). If A is an abelian category, we will denote by Proj-A ...

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Section: Preliminary Resultsmentioning

confidence: 99%

Modular perverse sheaves on flag varieties I: tilting and parity sheaves

Achar¹,

Simón²

2016

Ann. Sci. École Norm. Sup.

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“…For background on properly stratified categories, see [AR,§2]. In a properly stratified category-a notion that generalizes that of a quasihereditary category-there are four important classes of indecomposable objects, called standard, proper standard, costandard, and proper costandard objects.…”

Section: 2mentioning

confidence: 99%

“…Recall that PCoh(N ) is equipped with a recollement structure (see [AR,Proposition 2.2]). Let PCoh(N ) Γ ⊂ PCoh(N ) denote the Serre subcategory generated by∇(γ) m with γ ∈ Γ, and let Π Γ : PCoh(N ) → PCoh(N )/PCoh(N ) Γ be the Serre quotient functor.…”

Section: The Mirković-vilonen Conjecture For Mixed Sheavesmentioning

confidence: 99%

“…One of the main results of [ARc2] gives an equivalence of categories between parity sheaves on a flag variety and mixed tilting sheaves on the Langlands dual flag variety. Separately, according to [AR, Proposition 5.7], there is an equivalence of categorieswhere PCoh(N ) is the category of perverse-coherent sheaves on the nilpotent cone for G ∨ (see §2.6). These results raise the question of whether Parity (I) (Gr, k) participates in a "parity-tilting" equivalence.…”

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confidence: 99%

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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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