Global Langlands reciprocity

Global Langlands reciprocity is the conjectural principle that arithmetic global Langlands parameters for a reductive group $G$ organize automorphic representations of $G(\mathbb A_F)$ , with compatible localizations, $L$ -functions, and multiplicities.

It is a program of correspondences rather than one unconditional bijection.

Abelian case

For $G=\operatorname{GL}_1$ , global class field theory identifies characters of the idèle class group

F^\times\backslash\mathbb A_F^\times

with one-dimensional representations of the abelianized global Galois group, subject to the chosen reciprocity and Frobenius normalization. This is the abelian model for the general principle.

General linear groups

For $\operatorname{GL}_n$ , the expected primitive correspondence relates cuspidal automorphic representations to irreducible $n$ -dimensional global parameters. Over number fields this includes several different coefficient realizations and is known only in important families. Over global function fields, Drinfeld and Laurent Lafforgue proved the correspondence for $\operatorname{GL}_n$ , and Vincent Lafforgue constructed a broad automorphic-to-Galois parameterization for reductive groups.

General reductive groups

For a general $G$ , a parameter determines local packets rather than a single representation. A global packet is assembled from local members, and a multiplicity formula selects which restricted tensor products occur in the discrete spectrum. Endoscopy and Arthur parameters refine this statement.

Required compatibilities

The conjecture includes more than matching sets. It should preserve:

localization at every place;
unramified Satake parameters;
global and local $L$ - and epsilon factors;
central characters and duality;
functorial transfer under homomorphisms of $L$ -groups.

A result proving only one direction or only equality of almost-all unramified parameters should be named accordingly.

References

Robert P. Langlands, “Problems in the theory of automorphic forms,” in Lectures in Modern Analysis and Applications III, 1970. IAS copy.
Laurent Lafforgue, “Chtoucas de Drinfeld et correspondance de Langlands,” Inventiones Mathematicae 147 (2002), 1–241. DOI.
Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” JAMS 31 (2018), 719–891. arXiv.