Global Langlands parameter

Let $F$ be a global field and $G$ a connected reductive $F$ -group. In the conjectural Langlands formalism, a global Langlands parameter is an admissible homomorphism

\varphi:\mathcal L_F\longrightarrow {}^L G

from a global Langlands group $\mathcal L_F$ to the $L$ -group, considered up to conjugacy by $\widehat G$ . For every place $v$ , localization should produce a local $L$ -parameter $\varphi_v$ .

Number-field status

For a number field, the required group $\mathcal L_F$ has not been constructed in the generality demanded by the program. The display is therefore a conjectural organizing language, not an available definition of a concrete topological group.

Different realizations retain different parts of the expected parameter: complex representations of hypothetical Langlands groups, motivic Galois groups, and compatible systems of $\ell$ -adic Galois representations are related but are not interchangeable without hypotheses.

Function-field status

For a global function field, the absolute Galois group is concrete. Vincent Lafforgue's excursion operators attach semisimple $\widehat G$ -valued global Galois parameters to cuspidal automorphic data. This is a major theorem, but it does not turn the entire general packet and multiplicity formalism into a literal bijection.

Expected information

A global parameter should determine compatible local packets $\Pi_{\varphi_v}$ , global $L$ -functions, and a global packet inside the restricted product of the local packets. A multiplicity formula is then needed to decide which tensor products occur automorphically and with what multiplicity.

Tempered and Arthur parameters

A bounded global Langlands parameter is expected to describe tempered automorphic phenomena. The discrete spectrum also contains non-tempered representations, for which an Arthur parameter adds an $\operatorname{SL}_2(\mathbb C)$ -factor. The two parameter notions must not be conflated.

References

James Arthur, “A note on the automorphic Langlands group,” Canadian Mathematical Bulletin 45 (2002), 466–482. DOI.
Vincent Lafforgue, “Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale,” Journal of the American Mathematical Society 31 (2018), 719–891. arXiv.