The motivic Satake equivalence

Richarz, Timo; Scholbach, Jakob

doi:10.1007/s00208-021-02176-9

Cited by 7 publications

(7 citation statements)

References 45 publications

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“…Building upon our earlier work [RS20,RS19], the goal of the present paper is to refine (1.2) into an equivalence between the category of mixed Tate motives on the double quotient L + G\LG/L + G and representations of Deligne's extended dual group G 1 , a certain extension of G m (which records the weights) by G. This equivalence is independent of the choice of ℓ = p present in (1.2) via the use of ℓ-adic cohomology. We refer to Remark 5.8 for the relation with (1.1).…”

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confidence: 88%

Tate motives on Witt vector affine flag varieties

Richarz,

Scholbach

2020

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Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up. Contents 1. Introduction 1 2. Motives on perfect (ind-)schemes 4 3. Stratified Tate motives for perfect ind-schemes 10 4. Motives on Witt vector affine flag varieties 12 5. The motivic Satake equivalence 17 References 21

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confidence: 88%

Tate motives on Witt vector affine flag varieties

Richarz,

Scholbach

2020

Preprint

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“…Recall that there are motivic versions of the geometric Satake equivalence [90,75]. The above theorem can be regarded as their p-adic realization.…”

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confidence: 99%

Bessel $F$-isocrystals for reductive groups

Xu,

Zhu

2019

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We construct the Frobenius structure on a rigid connection Be Ǧ on Gm for a split reductive group Ǧ introduced by Frenkel-Gross. These data form a Ǧ-valued overconvergent F -isocrystal Be † Ǧ on G m,Fp , which is the p-adic companion of the Kloosterman Ǧ-local system Kl Ǧ constructed by Heinloth-Ngô-Yun. By studying the structure of the underlying differential equation, we calculate the monodromy group of Be † Ǧ when Ǧ is almost simple (which recovers the calculation of monodromy group of Kl Ǧ due to Katz and Heinloth-Ngô-Yun), and establish functoriality between different Kloosterman Ǧ-local systems as conjectured by Heinloth-Ngô-Yun. We show that the Frobenius Newton polygons of Kl Ǧ are generically ordinary for every Ǧ and are everywhere ordinary on |G m,Fp | when Ǧ is classical or G 2 . Contents 1. Introduction 1.1. Bessel equations and Kloosterman sums 1.2. Generalization for reductive groups 1.3. Strategy of the proof and the organization of the article 2. Review and complements on arithmetic D-modules 2.1. Overconvergent (F -)isocrystals and their rigid cohomologies 2.2. (Co)specialization morphism for de Rham and rigid cohomologies 2.3. Six functors formalism for arithmetic D-modules 2.4. Complements on the cohomology of arithmetic D-modules 2.5. Equivariant holonomic D-modules 2.6. Intermediate extension and the weight theory 2.7. Nearby and vanishing cycles 2.8. Universal local acyclicity 2.9. Local monodromy of an overconvergent F -isocrystal 2.10. Hyperbolic localization for arithmetic D-modules 3. Geometric Satake equivalence for arithmetic D-modules 3.1. The Satake category 3.2. Fusion product 3.3. Hypercohomology functor 3.4. Semi-infinite orbits 3.5. Tannakian structure and the Langlands dual group 3.6. The full Langlands dual group 4. Bessel F -isocrystals for reductive groups 4.1. Kloosterman F -isocrystals for reductive groups 4.2. Comparison between Kl dR Ǧ and Kl rig Ǧ 4.3. Comparison between Kl dR Ǧ and Be Ǧ 4.4. Bessel F -isocrystals for reductive groups 4.5. Monodromy groups

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“…This question was answered by T. Richarz and one of the authors (J.S.) for mixed Tate motives with rational coefficients in [RS21], and by X. Zhu for numerical motives in [Zhu18]. The goal of this paper is to generalize [RS21] by considering integral coefficients.…”

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confidence: 96%

“…for mixed Tate motives with rational coefficients in [RS21], and by X. Zhu for numerical motives in [Zhu18]. The goal of this paper is to generalize [RS21] by considering integral coefficients. Aside from addressing the above philosophical question, this has multiple concrete applications.…”

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confidence: 99%

“…One of our motivations is the ongoing project [RS20,RS21] on a motivic refinement of V. Lafforgue's global Langlands parametrization [Laf18]. Namely, the current work also refines [RS21] by not only considering mixed Tate motives on affine Grassmannians, but more generally on Beilinson-Drinfeld Grassmannians over powers of the affine line. The resulting factorization properties are a key tool in [Laf18].…”

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The geometric Satake equivalence for integral motives

Cass¹,

Hove²,

Scholbach³

2022

Preprint

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We prove the geometric Satake equivalence for mixed Tate motives over the integral motivic cohomology spectrum. This refines previous versions of the geometric Satake equivalence for split groups and power series affine Grassmannians. Our new geometric results include Whitney-Tate stratifications of Beilinson-Drinfeld Grassmannians and cellular decompositions of semi-infinite orbits. With future global applications in mind, we also achieve an equivalence relative to a power of the affine line. Finally, we use our equivalence to give Tannakian constructions of the C-group and a modified form of Vinberg's monoid.

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The motivic Satake equivalence

Abstract: We refine the geometric Satake equivalence due to Ginzburg, Beilinson–Drinfeld, and Mirković–Vilonen to an equivalence between mixed Tate motives on the double quotient $$L^+ G {\backslash }LG / L^+ G$$ L + G \ L G / L + G … Show more

Cited by 7 publications

References 45 publications

Tate motives on Witt vector affine flag varieties

Tate motives on Witt vector affine flag varieties

Bessel $F$-isocrystals for reductive groups

The geometric Satake equivalence for integral motives

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