Nilpotent Elements and Reductive Subgroups Over a Local Field

McNinch, George J.

doi:10.1007/s10468-020-10000-2

Cited by 6 publications

(27 citation statements)

References 28 publications

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“…Proof. The existence of x and ϕ follow from [McN,Theorem 4.5.2] and the arguments in [McN,Proof of Corollary 9.2.2]. Concerning uniqueness, since the morphism G(O) → G(F) is surjective (by [EGA4,Théorème 18.5.17]) we can assume that x ′ F = y.…”

Section: The Springer Resolutionmentioning

confidence: 95%

“…Concerning uniqueness, since the morphism G(O) → G(F) is surjective (by [EGA4,Théorème 18.5.17]) we can assume that x ′ F = y. In this case the claim is proved in [McN,Corollary 7.3.2]. The other important property we will use is the following.…”

Section: The Springer Resolutionmentioning

confidence: 97%

“…(Of course, Z G (x) K and Z G (x) K are also smooth.) Following [McN,Definition 1.4.1], a section x will be called balanced if…”

Section: The Springer Resolutionmentioning

confidence: 99%

“…We refer e.g. to [McN,Definition 3.2.1] or [J2,Definition 5.3] for the definition of a cocharacter associated with a nilpotent element. Recall also that under our assumptions there exists a canonical bijection from the set of nilpotent orbits of G K in g K to the set of nilpotent G F -orbits in g F , see [AHR1,§4.1] for references.…”

Section: The Springer Resolutionmentioning

confidence: 99%

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Integral exotic sheaves and the modular Lusztig-Vogan bijection

Achar,

Hardesty,

Riche

2018

Preprint

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Let G be a reductive group over an algebraically closed field k of very 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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Section: The Springer Resolutionmentioning

confidence: 95%

Section: The Springer Resolutionmentioning

confidence: 97%

“…(Of course, Z G (x) K and Z G (x) K are also smooth.) Following [McN,Definition 1.4.1], a section x will be called balanced if…”

Section: The Springer Resolutionmentioning

confidence: 99%

Section: The Springer Resolutionmentioning

confidence: 99%

See 2 more Smart Citations

Integral exotic sheaves and the modular Lusztig-Vogan bijection

Achar,

Hardesty,

Riche

2018

Preprint

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show abstract

“…We say that a section x is balanced if G x K K and G x F F are smooth group schemes and dim G x K K = dim G x F F (cf. [Mc3]).…”

Section: Introductionmentioning

confidence: 99%

On the centralizer of a balanced nilpotent section

Hardesty

2018

Preprint

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Let G be a split reductive algebraic group defined over a complete discrete valuation ring O, with residue field F and fraction field K, where the fiber G F is geometrically standard. A balanced nilpotent section x ∈ Lie(G) can roughly be thought of as an O-point in a K nilpotent orbit such that the corresponding orbits over K and F have the same Bala-Carter label. In this paper, we will establish a number of results on the structure of the centralizer G x ⊆ G of x. This includes a proof that G x is a smooth group scheme, and that the component groups of the geometric fibers G x K and G x F are isomorphic.Theorem 1.8. If x ∈ g O is a balanced nilpotent section, then there is a group isomorphism A(x K ) ∼ = A(x F ).Remark 1.9. This will be proven in §3.4.

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Nilpotent Orbits of Orthogonal Groups over p-adic Fields, and the DeBacker Parametrization

Bernstein

Nevins

et al. 2019

Algebr Represent Theor

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Nilpotent Elements and Reductive Subgroups Over a Local Field

Cited by 6 publications

References 28 publications

Integral exotic sheaves and the modular Lusztig-Vogan bijection

Integral exotic sheaves and the modular Lusztig-Vogan bijection

On the centralizer of a balanced nilpotent section

Nilpotent Orbits of Orthogonal Groups over p-adic Fields, and the DeBacker Parametrization

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