Kostant section, universal centralizer, and a modular derived Satake equivalence

Riche, Simon

doi:10.1007/s00209-016-1761-3

Cited by 16 publications

(50 citation statements)

References 33 publications

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“…We will also denote by I E S the restriction of I E to S E . Then by [R3,Corollary 3.5.8 & Proposition 4.5.3], I E S is a commutative smooth group scheme over S E . We denote by I E S its Lie algebra; it is a locally free sheaf of commutative O S E -Lie algebras (see [R3, §2.1]).…”

Section: Reminder On [R3]mentioning

confidence: 96%

“…The following result is proved in [R3,Propositions 3.5.5 & 4.5.2]. Here we consider the G m,E -action on t * E where x acts by multiplication by We denote by I E the universal centralizer associated with the action of G E on g E , see (2.1).…”

Section: Reminder On [R3]mentioning

confidence: 99%

“…By [R3,Lemma 4.2.3] there exists a G R -invariant symmetric bilinear form on g R which is a perfect pairing. We fix once and for all such a bilinear form, and we denote by κ : g R ∼ − → g * R the induced (G R -equivariant) isomorphism.…”

Section: 2mentioning

confidence: 99%

“…With these definitions, κ is G m,Requivariant, e is fixed by the action, and b + R and n + R are G m,R -stable. By [R3,Lemma 4.3.1]…”

Section: Reminder On [R3]mentioning

confidence: 99%

“…For the remainder of this subsection, let us consider the case E = F. First we recall that there exists a unique smooth commutative group scheme J F over t * F whose pull-back under ν reg is the restriction I F reg of I F to g reg F , see [R3,Proposition 3.3.9]. Now by [R3,Remark 3.5.9], the composition…”

Section: Reminder On [R3]mentioning

confidence: 99%

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Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

Mautner

Riche

2018

J. Eur. Math. Soc.

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Let G be a connected reductive group over an algebraically closed field F of good characteristic, satisfying some mild conditions. In this paper we relate tilting objects in the heart of Bezrukavnikov's exotic t-structure on the derived category of equivariant coherent sheaves on the Springer resolution of G, and Iwahori-constructible F-parity sheaves on the affine Grassmannian of the Langlands dual group. As applications we deduce in particular the missing piece for the proof of the Mirković-Vilonen conjecture in full generality (i.e. for good characteristic), a modular version of an equivalence of categories due to Arkhipov-Bezrukavnikov-Ginzburg, and an extension of this equivalence./ / Tilt(E G×Gm ( N )) commutes. Hence to conclude we only have to prove that the diagram

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Section: Reminder On [R3]mentioning

confidence: 96%

Section: Reminder On [R3]mentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

“…With these definitions, κ is G m,Requivariant, e is fixed by the action, and b + R and n + R are G m,R -stable. By [R3,Lemma 4.3.1]…”

Section: Reminder On [R3]mentioning

confidence: 99%

Section: Reminder On [R3]mentioning

confidence: 99%

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Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

Mautner

Riche

2018

J. Eur. Math. Soc.

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On the Humphreys Conjecture on Support Varieties of Tilting Modules

Achar

Hardesty

Riche

2019

Transformation Groups

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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.

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On two geometric realizations of an affine Hecke algebra

Bezrukavnikov

2015

Publ.math.IHES

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Abstract. The article is a contribution to the local theory of geometric Langlands duality. The main result is a categorification of the isomorphism between the (extended) affine Hecke algebra associated to a reductive group G and Grothendieck group of equivariant coherent sheaves on Steinberg variety of Langlands dual group Gˇ; this isomorphism due to Kazhdan-Lusztig and Ginzburg is a key step in the proof of tamely ramified local Langlands conjectures.The paper is a continuation of [1], [12], it relies on technical material developed in [23].

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Kostant section, universal centralizer, and a modular derived Satake equivalence

Cited by 16 publications

References 33 publications

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

On the Humphreys Conjecture on Support Varieties of Tilting Modules

On two geometric realizations of an affine Hecke algebra

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