Character sheaves on unipotent groups in positive characteristic: foundations

Boyarchenko, Mitya; Drinfeld, Vladimir

doi:10.1007/s00029-013-0133-7

Cited by 28 publications

(84 citation statements)

References 41 publications

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“…Thus, (Cone(ε), P 1 n ) forms a pair of complementary idempotents in K − (SBim n ), in the sense of [15]. Many properties of P 1 n can be deduced immediately from the axioms together with some basic theory of categorical idempotents developed in [15] (also [3]). For instance:…”

Section: Axiomaticsmentioning

confidence: 99%

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Categorified Young symmetrizers and stable homology of torus links II

Abel

Hogancamp

2017

Sel. Math. New Ser.

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We construct complexes P 1 n of Soergel bimodules which categorify the Young idempotents corresponding to one-column partitions. A beautiful recent conjecture [12] of Gorsky-Rasmussen relates the Hochschild homology of categorified Young idempotents with the flag Hilbert scheme. We prove this conjecture for P 1 n and its twisted variants. We also show that this homology is also a certain limit of Khovanov-Rozansky homologies of torus links. Along the way we obtain several combinatorial results which could be of independent interest. Definition 1.1. Let R → P ∨ 1 n denote an injective resolution of R, thought of as a graded R ⊗ R W R-module.We note for future reference that R ⊗ R W R is self-injective. The complex P ∨ 1 n is graded infinite, being supported in all non-negative homological degrees. This complex categorifies the one-column Young symmetrizer in a sense explained below.Let Ch(R-mod-R) denote the category of complexes of finitely generated, graded (R, R)-bimodules. Our first main theorem says that P ∨ 1 n is related to a certain braid group action on Ch(R-mod-R). Associated to each n-strand braid,denote the Rouquier complex associated to the (positive) full twist braid. Then {FT ⊗k } ∞ k=0 , after the appropriate grading shifts, can be made into a direct system with homotopy colimit P ∨ 1 n . Remark 1.3. Dually, one may consider a complex P 1 n which is defined to be a projective resolution P 1 n → R viewed as a graded R ⊗ R W R-module. Analogously, if we let FT −1 denote the Rouquier complex of the negative full twist braid then {(FT −1 ) ⊗k } ∞ k=0 can made into an inverse system with homotopy limit P 1 n , after the appropriate grading shifts. We will prove in §2.6 that the duality functor (·) ∨ : Ch(R-mod-R) → Ch(R-mod-R) sends P 1 n to P ∨ 1 n . Our notation prefers complexes which are bounded above, in order to be consistent with [15]. However, in this paper the complex P ∨ 1 n is often nicer than P 1 n , since P ∨ 1 n is an algebra in the homotopy category K + (R-mod-R), while P 1 n is a coalgebra. This is discussed in Remark 1.13. The algebra structure makes writing our following main results easier. See Example 4.13 for a sample computation in the non-dual, n = 2 case.Let us recall the construction of Khovanov and Rozansky's triply-graded link homology given in [17]. Hochschild cohomology of bimodules defines a functor HHH from Ch(R-mod-R) to the category of triply-graded vector spaces, and HHH(F (β)) is a well-defined invariant of the oriented linkβ, up to isomorphism and overall shift of triply-graded vector spaces. This invariant is called Khovanov-Rozansky homology, hereafter referred to as KR homology.Remark 1.4. Actually, in [17] the results are stated in terms of Hochschild homology. In this paper, as in [15], we prefer Hochschild cohomology to Hochschild homology; there is not much difference since for polynomial rings the two are isomorphic up to regrading. Our convention will ensure that HHH applied to our projector is a graded commutative algebra, rather than an algebra up to regrad...

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Section: Axiomaticsmentioning

confidence: 99%

“…For more of an explanation of the relation between Hochschild cohomology of P ∨ 1 n and its ring of endomorphisms, see §4. 3. The special polynomials a ij (x, y) are defined in Definition 2.36, and are essentially the double Schubert polynomials S w (x, y) (See [19] for more information).…”

Section: Introductionmentioning

confidence: 99%

Categorified Young symmetrizers and stable homology of torus links II

Abel

Hogancamp

2017

Sel. Math. New Ser.

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“…In this paper, by passing to the perfectizations, we will always assume that all our groups are in fact perfect quasi-algebraic groups over k even though we may not mention this explicitly. We refer to [BD,§1.9] for more about this convention. With this convention, the Frobenius maps in fact become automorphisms.…”

Section: Preliminary Constructionsmentioning

confidence: 99%

Shintani descent for algebraic groups and almost characters of unipotent groups

Deshpande

2016

Compositio Math.

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In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field Fq. For this, it is essential to treat all the pure inner Fq-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the "trace of Frobenius" functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko-Drinfeld's theory of character sheaves on neutrally unipotent are in fact positive integral confirming a conjecture due to Drinfeld.

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“…Our aim in this paper is to present a detailed and elementary construction of the inverse perfection of an F p -scheme and discuss some of its properties. The (inverse) perfection functor has played, a continues to play, a significant role in algebraic geometry (see, for example, [Ser1,Ser2,BD,BW,Pep,KL]). We believe that our presentation will be useful to all students and researchers that at some point in their studies will need to consider the (inverse) perfection of an F p -scheme.…”

Section: Introductionmentioning

confidence: 99%