On exotic and perverse-coherent sheaves

2019

Transformation Groups

Self Cite

Let G be a simply-connected semisimple algebraic group over an algebraically closed field of characteristic p, assumed to be larger than the Coxeter number. The "support variety" of a G-module M is a certain closed subvariety of the nilpotent cone of G, defined in terms of cohomology for the first Frobenius kernel G 1 . In the 1990s, Humphreys proposed a conjectural description of the support varieties of tilting modules; this conjecture has been proved for G = SLn in earlier work of the second author.In this paper, we show that for any G, the support variety of a tilting module always contains the variety predicted by Humphreys, and that they coincide (i.e., the Humphreys conjecture is true) when p is sufficiently large. We also prove variants of these statements involving "relative support varieties."1.2. The Humphreys conjecture. We fix a Borel subgroup B ⊂ G and a maximal torus T ⊂ G, denote by X the character lattice of T , and let X + ⊂ X be the subset of dominant weights (for the choice of positive roots such that B is the P.A.

Section: Support Computationsmentioning

confidence: 99%

On the Humphreys Conjecture on Support Varieties of Tilting Modules

Achar

Hardesty

2019

Transformation Groups

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“…Following the notation and conventions of [7, §9], we fix a subset I ⊂ S, and consider the objects ∆ I (λ) and ∇ I (λ) in D b CohĠ ×G m ( N I ) characterized in [7,Proposition 9.16]. 1 Here λ ∈ X +,reg I where X +,reg I := {λ ∈ X | ∀s ∈ I, λ, α ∨ s > 0}. In order to define these objects one needs to choose an order ≤ on X.…”

Section: B a Graded Exceptional Setmentioning

confidence: 99%

“…In particular, this theory provides standard and costandard (mixed) perverse sheaves 2 ∆ mix w and ∇ mix w for all w ∈ 0 W aff , and indecomposable tilting perverse sheaves T mix w . We will denote by {1} the autoequivalence of D mix (Iw) (Gr , k) induced by the cohomological shift in Parity (Iw) (Gr , k), and by [1] the usual shift of complexes; then the "Tate twist" 1 := {−1} [1] is t-exact for the perverse t-structure.…”

Section: C Mixed Derived Category Of the Affine Grassmannianmentioning

confidence: 99%

“…The exotic t-structure is a remarkable t-structure on this category, originally defined in [14]. This t-structure has close connections to derived equivalences coming from the geometric Langlands program [9,8,23], to the cohomology of tilting modules for Lusztig's quantum groups [14] and algebraic groups [2], and other topics in representation theory (see [1] and the references therein). It is defined using a so-called exceptional set of objects in D b CohĠ ×G m ( N ).…”

Section: Introduction 1a the Exotic T-structurementioning

confidence: 99%

See 1 more Smart Citation

The parabolic exotic t-structure

Achar¹,

Cooney

Épijournal De Géométrie Algébrique

2018

International audience Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks. Soit G un groupe algébrique réductif connexe sur un corps k algébriquement clos. La t-structure exotique sur le fibré cotangent de sa variété de drapeaux T^*(G/B), introduite à l'origine par Bezrukavnikov, a été un outil clé pour de nombreux résultats majeurs en théorie géométrique des représentations, en particulier la démonstration de la conjecture de Finkelberg-Mirkovic graduée. Dans cet article, nous étudions (sous de légères hypothèses techniques) une t-structure analogue sur le fibré cotangent de la variété de drapeaux partiels T^*(G/P). Comme application, nous prouvons un analogue parabolique de l'équivalence de Arkhipov-Bezrukavnikov-Ginzburg. Lorsque la caractéristique de k est supérieure au nombre de Coxeter, nous déduisons un analogue de la conjecture de Finkelberg-Mirkovic graduée pour certains blocs singuliers.

“…of the form J T −1 w (O N ) m , for w ∈ W aff and m ∈ Z. It is convenient (see [A2,MR]) to use a slightly different normalization of the standard and costandard objects, setting…”

Section: Affine Braid Groupmentioning

confidence: 99%

On the Exotic t-Structure in Positive Characteristic

Mautner

2015

Int Math Res Notices

104

Abstract. In this paper we study Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of a connected reductive algebraic group defined over a field of positive characteristic with simply-connected derived subgroup. In particular, we show that the heart of the exotic t-structure is a graded highest weight category, and we study the tilting objects in this heart. Our main tool is the "geometric braid group action" studied by Bezrukavnikov and the second author.