Silting complexes of coherent sheaves and the Humphreys conjecture

Achar, Pramod N.; Hardesty, William

doi:10.48550/arxiv.2106.04268

Cited by 2 publications

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“…Since this paper first appeared in preprint form, the authors have obtained [AH2] a proof of the (set-theoretic) relative Humphreys conjecture CO-t-STRUCTURES ON DERIVED CATEGORIES 51 in general, by an argument that makes crucial use of the co-t-structure machinery developed in this paper. (However, the scheme-theoretic version proposed in Section 7 remains open outside of GL n .…”

Section: Added In Revisionmentioning

confidence: 99%

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2024

Represent. Theory

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We construct a co-t-structure on the derived category of coherent sheaves on the nilpotent cone N of a reductive group, as well as on the derived category of coherent sheaves on any parabolic Springer resolution. These structures are employed to show that the push-forwards of the "exotic parity objects" (considered by Achar, Hardesty, and Riche [Transform. Groups 24 (2019), pp. 597-657]), along the (classical) Springer resolution, give indecomposable objects inside the coheart of the co-t-structure on N . We also demonstrate how the various parabolic co-t-structures can be related by introducing an analogue to the usual translation functors. As an application, we give a proof of a scheme-theoretic formulation of the relative Humphreys conjecture on support varieties of tilting modules in type A for p > h.

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Section: Added In Revisionmentioning

confidence: 99%

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2024

Represent. Theory

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“…To avoid confusion we use subscripts to specify the base ring of schemes. (However, we do not use subscripts for morphisms, since the ring under consideration † After this paper appeared in preprint form, a proof of the relative version of the Humphreys conjecture for 𝑝 larger than the Coxeter number was obtained by the first two authors [6]. ‡ Some of these conjectures have now been proved in [6].…”

Section: Conventionmentioning

confidence: 99%

Integral exotic sheaves and the modular Lusztig–Vogan bijection

Achar

Hardesty

Riche

2022

Journal of London Math Soc

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Let 𝐺 be a reductive algebraic group over an algebraically closed field 𝕜 of pretty good characteristic. The Lusztig-Vogan bijection is a bijection between the set of dominant weights for 𝐺 and the set of irreducible 𝐺-equivariant vector bundles on nilpotent orbits, conjectured by Lusztig and Vogan independently, and constructed in full generality by Bezrukavnikov. In characteristic 0, this bijection is related to the theory of 2-sided cells in the affine Weyl group, and plays a key role in the proof of the Humphreys conjecture on support varieties of tilting modules for quantum groups at a root of unity. In this paper, we prove that the Lusztig-Vogan bijection is (in a way made precise in the body of the paper) independent of the characteristic of 𝕜. This allows us to extend all of its known properties from the characteristic-0 setting to the general case. We also expect this result to be a step towards a proof of the Humphreys conjecture on support varieties of tilting modules for reductive groups in positive characteristic.

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Silting complexes of coherent sheaves and the Humphreys conjecture

Cited by 2 publications

References 16 publications

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Integral exotic sheaves and the modular Lusztig–Vogan bijection

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