The integral geometric Satake equivalence in mixed characteristic

Yu, Jize

doi:10.48550/arxiv.1903.11132

Cited by 3 publications

(9 citation statements)

References 8 publications

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“…Using a degeneration of the local Hecke stack, which is essentially the B + dRaffine Grassmannian of [SW20], to the Witt vector affine Grassmannian, Theorem I.2.12 gives a new proof of Zhu's geometric Satake equivalence for the Witt vector affine Grassmannian [Zhu17]. In fact, we even prove a version with Z -coefficients, thus also recovering the result of Yu [Yu19].…”

Section: I2 the Big Picturementioning

confidence: 59%

“…G,k ; this was considered by Zhu [Zhu17] and Yu [Yu19]. In particular, this discussion implies the following result that we will need later.…”

Section: Moreover Filtering By Cohomology Sheaves We Can Actually Ass...mentioning

confidence: 74%

“…is fully faithful. If d = 1 and S = Spd F q , then one can compare it to the category considered by Zhu [Zhu17] and Yu [Yu19], defined in terms of the Witt vector affine Grassmannian. Moreover, the categories for S = Spd O C and S = Spd C are naturally equivalent to the category for S = Spd F q , via the base change functors; here C = E. Thus, the Satake category is, after picking a reductive model of G, naturally the same for the Witt vector affine Grassmannian and the B + dR -affine Grassmannian.…”

Section: Geometric Satakementioning

confidence: 99%

“…Moreover, the categories for S = Spd O C and S = Spd C are naturally equivalent to the category for S = Spd F q , via the base change functors; here C = E. Thus, the Satake category is, after picking a reductive model of G, naturally the same for the Witt vector affine Grassmannian and the B + dR -affine Grassmannian. At this point, we could in principle use Zhu's results [Zhu17] (refined integrally by Yu [Yu19]) to identify the Satake category with the category of representations of G, at least when G is unramified. However, for the applications we actually need finer knowledge of the functoriality of the Satake equivalence including the case for d > 1; we thus prove everything we need directly.…”

Section: Geometric Satakementioning

confidence: 99%

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Geometrization of the local Langlands correspondence

Fargues,

Scholze

2021

Preprint

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Following the idea of [Far16], we develop the foundations of the geometric Langlands program on the Fargues-Fontaine curve. In particular, we define a category of -adic sheaves on the stack BunG of G-bundles on the Fargues-Fontaine curve, prove a geometric Satake equivalence over the Fargues-Fontaine curve, and study the stack of L-parameters. As applications, we prove finiteness results for the cohomology of local Shimura varieties and general moduli spaces of local shtukas, and define L-parameters associated with irreducible smooth representations of G(E), a map from the spectral Bernstein center to the Bernstein center, and the spectral action of the category of perfect complexes on the stack of L-parameters on the category of -adic sheaves on BunG. ContentsChapter I. Introduction I.1. The local Langlands correspondence I.2. The big picture I.3. The Fargues-Fontaine curve I.4. The geometry of Bun G I.5. -adic sheaves on Bun G I.6. The geometric Satake equivalence I.7. Cohomology of moduli spaces of shtuka I.8. The stack of L-parameters I.9. Construction of L-parameters I.10. The spectral action I.11. The origin of the ideas I.12. Acknowledgments I.13. Notation Chapter II. The Fargues-Fontaine curve and vector bundles V.1. Classifying stacks V.2. Étale sheaves on strata V.3. Local charts V.4. Compact generation V.5. Bernstein-Zelevinsky duality V.6. Verdier duality V.7. ULA sheaves Chapter VI. Geometric Satake VI.1. The Beilinson-Drinfeld Grassmannian VI.2. Schubert varieties VI.3. Semi-infinite orbits VI.4. Equivariant sheaves VI.5. Affine flag variety VI.6. ULA sheaves VI.7. Perverse Sheaves VI.8. Convolution VI.9. Fusion VI.10. Tannakian reconstruction VI.11. Identification of the dual group VI.12. Cartan involution Chapter VII. D (X) VII.1. Solid sheaves VII.2. Four functors VII.3. Relative homology VII.4. Relation to D ét VII.5. Dualizability VII.6. Lisse sheaves VII.7. D lis (Bun G ) Chapter VIII. L-parameter VIII.1. The stack of L-parameters VIII.2. The singularities of the moduli space VIII.3. The coarse moduli space VIII.4. Excursion operators VIII.5. Modular representation theory Chapter IX. The Hecke action IX.1. Condensed ∞-categories IX.2. Hecke operators IX.3. Cohomology of local Shimura varieties IX.4. L-parameter IX.5. The Bernstein center IX.6. Properties of the correspondence IX.7. Applications to representations of G(E) CONTENTS Chapter X. The spectral action X.1. Rational coefficients X.2. Elliptic parameters X.3. Integral coefficients Bibliography CHAPTER I

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Section: I2 the Big Picturementioning

confidence: 59%

“…G,k ; this was considered by Zhu [Zhu17] and Yu [Yu19]. In particular, this discussion implies the following result that we will need later.…”

Section: Moreover Filtering By Cohomology Sheaves We Can Actually Ass...mentioning

confidence: 74%

Section: Geometric Satakementioning

confidence: 99%

Section: Geometric Satakementioning

confidence: 99%

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Geometrization of the local Langlands correspondence

Fargues,

Scholze

2021

Preprint

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“…We briefly recall the geometric properties of the mixed characteristic affine Grassmannians and the integral coefficient geometric Satake equivalence. We refer to [19] and [17] for more details.…”

Section: The Integral Coefficient Geometric Satake Equivalence In Mix...mentioning

confidence: 99%

A Geometric Jacquet-Langlands Transfer for automorphic forms of higher weights

Yu¹

2020

Preprint

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In this paper, we give a geometric construction of the Jacquet-Langlands transfer for automorphic forms of higher weights by studying the geometry of the mod p fibres of different Hodge type Shimura varieties which satisfy a mild assumption and the cohomological correspondences between them. Contents 1. Introduction 1.1. Main theorem 1.2. Plan of the paper 1.3. Notations 1.4. Acknowledgements 2. The Integral Coefficient Geometric Satake Equivalence in Mixed Characteristic 2.1. The geometry of mixed characteristic affine Grassmannians 2.2. The Satake category 2.3. More on the convolution Grassmannians 2.4. The geometric Satake equivalence 3. Local Hecke Stacks 3.1. Torsors over the local Hecke stacks 3.2. Perverse sheaves on the moduli of local Hecke stacks 4. Moduli of Local Shtukas 4.1. Moduli of restricted local Shtukas 4.2. Perverse sheaves on the moduli of local Shtukas 5. Key theorem 5.1. Set up 5.2. More on local Hecke stacks 5.3. More on moduli of local Shtukas 5.4. The category of coherent sheaves on the stack of unramified Langlands parameters 5.5. Key theorem 6. Cohomological correspondences between Shimura varieties 6.1. Preliminaries 6.2. Étale local systems 6.3. Main theorem 6.4. Non-vanishing of the geometric Jacquet-Langlands transfer References

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On the Kottwitz conjecture for local shtuka spaces

Kaletha

Hansen

Weinstein³

2022

Forum of Mathematics, Pi

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Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of $\operatorname {\mathrm {GL}}_n$ , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.

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The integral geometric Satake equivalence in mixed characteristic

Cited by 3 publications

References 8 publications

Geometrization of the local Langlands correspondence

Geometrization of the local Langlands correspondence

A Geometric Jacquet-Langlands Transfer for automorphic forms of higher weights

On the Kottwitz conjecture for local shtuka spaces

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