The affine Grassmannian and the Springer resolution in positive characteristic

Hardesty

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: Exotic Parity Complexesmentioning

confidence: 99%

“…for any λ ∈ X (see [MR2,Theorem 1.2] or [ARd,Theorem 8.3]). The indecomposable tilting objects in Perv mix (Iw) (Gr, k) are parametrized in a natural way by X × Z; we denote by T (λ) the indecomposable object associated with (λ, 0) (so that the indecomposable object associated with (λ, n) is T (λ) n ).…”

Section: Exotic Parity Complexesmentioning

confidence: 99%

On the Humphreys Conjecture on Support Varieties of Tilting Modules

Hardesty

2019

Transformation Groups

Self Cite

“…Note that, since we already know that F I is a degrading functor, R Ind G PI is fully-faithful if and only if R Ind G PI • F I is a degrading functor. Moreover, in the special case I = ∅, there is a rich supply of objects with favorable Ext-properties in D b CohĠ ×Gm ( N ∅ ): namely, the standard and costandard objects in the heart of the exotic t-structure, which has been introduced by Bezrukavnikov [14] and studied further in [6,43]. Using the special case I = {s} in Figure 2, we prove in Section 10 that R Ind G B • F ∅ takes standard (resp.…”

Section: Introductionmentioning

confidence: 99%

Reductive groups, the loop Grassmannian, and the Springer resolution

2018

Invent. math.

Self Cite

In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a "graded version" of a conjecture of Finkelberg-Mirković describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian, and deduce a new proof of Lusztig's conjecture in large characteristic. Part 2. Formality theorems 34 5. Formality for P I,1 -modules 34 6.Ṗ J -equivariant formality 41 7. Compatibility with induction 49 Part 3. Induction theorems 58 8. Translation functors 58 9. Cotangent bundles of partial flag varieties 75 10. The induction theorem 92 11. The graded Finkelberg-Mirković conjecture 97 Index of notation 109 References 110 P.A.

“…graded Finkelberg-Mirković conjecture [AR4] parabolic-Whittaker duality monoidal Koszul duality [AR4] [ ARd2,MR] tilting character formula (conjectured in [RW]) Figure 1.1. Reductive groups and Koszul duality C-sheaves on G/B which are constructible with respect to the stratification by Borbits (called the Bruhat stratification), and let Perv (B) (G/B, C) be the heart of the perverse t-structure on this category.…”

mentioning

confidence: 99%

Koszul duality for Kac–Moody groups and characters of tilting modules

Makisumi

et al. 2018

J. Amer. Math. Soc.

We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic ℓ in terms of ℓ-Kazhdan-Lusztig polynomials, for ℓ > h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if ℓ ≥ 2h − 2. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.1.3. The Kac-Moody case and quantum groups. These ideas were later generalized by Bezrukavnikov-Yun [BY] to the case where G is replaced by a general Kac-Moody group G . Let B ⊂ G be a Borel subgroup, and let U ⊂ B be its unipotent radical. An important new idea in [BY] (also suggested in [BG]) is that a richer version of Koszul duality can be obtained if one "deforms" the categories of semisimple complexes on G /B and tilting perverse sheaves on G ∨ /B ∨ along a polynomial ring. The B-constructible semisimple complexes are thus replaced by the B-equivariant semisimple complexes, and the tilting perverse sheaves are replaced by the so-called "free-monodromic" objects constructed (via a very technical procedure) by Yun using certain pro-objects in the derived category of G ∨ /U ∨ , see [BY, Appendix A]. These deformed categories each have a monoidal structure, given by an appropriate kind of convolution product. The main result of [BY] is an equivalence of monoidal categoriesrelating B-equivariant semisimple complexes on G /B and free-monodromic tilting perverse sheaves attached to G ∨ . From this, Bezrukavnikov-Yun then deduce a Kac-Moody analogue of (1.1). As in §1.2, this result has a combinatorial motivation in terms of Kazhdan-Lusztig polynomials [Yu], and a representation-theoretic motivation in terms of analogues of the category O for Kac-Moody Lie algebras.But a third motivation for the work in [BY], specifically in the case of affine Kac-Moody groups, came from the hope of uniting two geometric approaches to the study of representations of Lusztig's quantum groups at a root of unity (see e.g. [Be, §1.2]), which we review below. Let Rep 0 (U ζ ) denote the principal block of the category of finite-dimensional representations of Lusztig's quantum group U ζ associated with an adjoint semisimple complex algebraic group G, specialized at a root of unity ζ.The first approach comes from [ABG]. The main result of [ABG, Part I] relates 3 Rep 0 (U ζ ) to the derived category of equivariant coherent sheaves on the Springer resolution N of G, denoted by D b Coh G×Gm ( N ). Then the main result of [ABG, Part II] states that D b Coh G×Gm ( N ) is equivalent to the derived category of Iwahoriconstructible perverse sheaves on the affine Grassmannian Gr of the Langlands dual semisimple group G ∨ . Together, these results give a new proof of Lusztig's character formula for simple modules in Rep 0 (U ζ ). (This character formula was already known when [ABG] appeared, by combining work of Kazhdan-Lusztig [KL2], Lusztig [Lu2] and Kashiwara-Tanisaki [KT].)For each s ∈ J, choose, once and for all, a...