Shutkas for reductive groups and global Langlands parametrization. (Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale.) (French. English summary) Zbl 1395.14017
In this article the global Langlands correspondence for reductive groups over a function field is proved in the direction from automorphic representations to Galois representations. Let \(X\) be a smooth projective curve over \({\mathbb F}_q\), geometrically irreducible, with function field \(F\). Let \(G\) be a connected reductive group over \(F\). Considered are cusp forms on \(G({\mathbb A})\) with values in a finite extension of \({\mathbb Q}_l\).
Using the cohomology of \(G\)-shtukas the author constructs “excursion” operators on spaces of cusp forms, commuting with Hecke operators. Considering commutative algebras generated by excursion operators one obtains a decomposition of the space of cusp forms. An application of the geometric Satake equivalence makes the decomposition canonical. This decomposition is indexed by certain \(\hat {G}\)-conjugacy classes of morphisms \(\text{Gal}(\bar{F}/F)\rightarrow \,^L\!G\), which are proved to be the Langlands parameters (\(\hat{G}\) and \(^L\!G\) are taken over \(\bar{{\mathbb Q}}_l\)).
In the introduction the proof is sketched under the assumption that \(G\) is split over \({\mathbb F}_q\) (30 pages). The complete proof also is first given for split \(G\) and after that the changes to be made in the general case are discussed.
The proof is independent of the Arthur-Selberg trace formula.
Using the cohomology of \(G\)-shtukas the author constructs “excursion” operators on spaces of cusp forms, commuting with Hecke operators. Considering commutative algebras generated by excursion operators one obtains a decomposition of the space of cusp forms. An application of the geometric Satake equivalence makes the decomposition canonical. This decomposition is indexed by certain \(\hat {G}\)-conjugacy classes of morphisms \(\text{Gal}(\bar{F}/F)\rightarrow \,^L\!G\), which are proved to be the Langlands parameters (\(\hat{G}\) and \(^L\!G\) are taken over \(\bar{{\mathbb Q}}_l\)).
In the introduction the proof is sketched under the assumption that \(G\) is split over \({\mathbb F}_q\) (30 pages). The complete proof also is first given for split \(G\) and after that the changes to be made in the general case are discussed.
The proof is independent of the Arthur-Selberg trace formula.
Reviewer: J. G. M. Mars (Utrecht)
MSC:
| 14G35 | Modular and Shimura varieties |
| 14H60 | Vector bundles on curves and their moduli |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
| 22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |
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