Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

Lafforgue, Vincent

doi:10.1142/9789813272880_0025

Cited by 4 publications

(5 citation statements)

References 52 publications

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“…Remark 3.9. There are precise conjectures [LafICM,(8.3)] to the effect that the following is always true:…”

Section: The Main Theoremsmentioning

confidence: 99%

On the generalized Ramanujan conjecture over function fields

Ciubotaru¹,

Harris²

2022

Preprint

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Let G be a simple group over a global function field K, and let π be a cuspidal automorphic representation of G. Suppose K has two places u and v (satisfying a mild restriction on the residue field cardinality), at which the group G is quasi-split, such that πu is tempered and πv is unramified and generic. We prove that π is tempered at all unramified places Kw at which G is unramified quasi-split.The proof uses the Galois parametrization of cuspidal representations due to V. Lafforgue to relate the local Satake parameters of π to Deligne's theory of Frobenius weights. The main observation is that, in view of the classification of generic unitary spherical representations, due to Barbasch and the first-named author, the theory of weights excludes generic complementary series as possible local components of π. This in turn determines the local Frobenius weights at all unramified places. In order to apply this observation in practice we need a result of the second-named author with Gan and Sawin on the weights of discrete series representations.2 Laurent Clozel has pointed out that, in his review of Shahidi's article for Mathematical Reviews, Wee Teck Gan noted that the version of the Arthur Conjectures assumed in Shahidi's Theorem 6.2 includes the Ramanujan Conjecture for GL(n). For number fields this is quite far from being established, but for function fields this is due to Laurent Lafforgue [Laf02, Théorème VI.10].3 Here and elsewhere, we use this expression as shorthand for the condition that G, as an algebraic group over Kv, is unramified and quasi-split.

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“…Remark 3.9. There are precise conjectures [LafICM,(8.3)] to the effect that the following is always true:…”

Section: The Main Theoremsmentioning

confidence: 99%

On the generalized Ramanujan conjecture over function fields

Ciubotaru¹,

Harris²

2022

Preprint

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“…Remark 16.1.9. We propose LocSys arithm G (X) as a candidate for the stack S, alluded to in [VLaf2,Remark 8.5].…”

Section: Denotementioning

confidence: 99%

“…Remark 16.4.7. As has been remarked above, we propose our LocSys arithm Ǧ (X) as a candidate for the stack sought-for in [VLaf2,Remark 8.5] and [LafZh,Sect. 6] (it was denoted S/ Ǧ in both these papers).…”

Section: Denotementioning

confidence: 99%

The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support

Arinkin¹,

Gaitsgory²,

Kazhdan³

et al. 2020

Preprint

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Introduction 0.1. Starting point 0.2. Summary 0.3. Some antecedents 0.4. Contents 0.5. Overview: the stack LocSys restr G (X) 0.6. Overview: spectral decomposition 0.7. Overview: Shv Nilp (BunG) 0.8. Overview: Langlands theory 0.9. Notations and conventions 0.10. Acknowledgements 1. The restricted version of the stack of local systems 1.1. Lisse sheaves 1.2. Another version of lisse sheaves 1.3. Definition of LocSys restr G (X) as a functor 1.4. Rigidification 1.5. Convergence 1.6. Deformation theory 1.7. A Tannakian intervention 1.8. Proof of ind-representability 2. Uniformization and the end of proof of Theorem 1.3.2 2.1. What is there left to prove? 2.2. Uniformization 2.3. Dominance of the uniformization morphism 2.4. Analysis of connected/irreducible components 3. Comparison with the Betti and de Rham versions of LocSys G (X) 3.1. Relation to the Rham version 3.2. A digression: ind-closed embeddings 3.3. Uniformization and the proof of Theorem 3.1.7 3.4. The Betti version of LocSys G (X) 3.5. Relationship of the restricted and Betti versions 3.6. The coarse moduli space of homomorphisms 4. The formal coarse moduli space 4.1. The "mock-properness" of red LocSys restr G 4.2. A digression: ind-algebraic stacks 4.3. Mock-affineness and coarse moduli spaces 4.4. Coarse moduli spaces for connected components of LocSys restr G (X)

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“…The first derives from the work of V. Lafforgue on the global Langlands correspondence over function fields [La18a,La18b]. Lafforgues' construction in Drinfeld's interpretation (cf.…”

Section: Thus Informallymentioning

confidence: 99%

Coherent Springer theory and the categorical Deligne-Langlands correspondence

Ben‐Zvi¹,

Chen²,

Helm³

et al. 2020

Preprint

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Kazhdan and Lusztig identified the affine Hecke algebra H with an equivariant Kgroup of the Steinberg variety, and applied this to prove the the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations with an Iwahori-fixed vector of reductive groups over nonarchimedean local fields. We apply techniques from derived algebraic geometry to pass from K-theory to Hochschild homology and thereby identify H with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of H-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Hellmann and Zhu). We explain how this refines the more familiar description of representations, one central character at a time, in terms of categories of perverse sheaves (as previously observed in local Langlands over R).In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of GLnpKq into coherent sheaves on the stack of Langlands parameters.

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Shtukas for Reductive Groups and Langlands Correspondence for Function Fields

Cited by 4 publications

References 52 publications

On the generalized Ramanujan conjecture over function fields

On the generalized Ramanujan conjecture over function fields

The stack of local systems with restricted variation and geometric Langlands theory with nilpotent singular support

Coherent Springer theory and the categorical Deligne-Langlands correspondence

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