Basic local Langlands correspondence

Let $F$ be a local field and $G$ a connected reductive $F$ -group. Write $\operatorname{Irr}(G(F))$ for the equivalence classes of irreducible admissible representations of $G(F)$ , where admissibility has its nonarchimedean or archimedean meaning according to $F$ , and write $\Phi(G)$ for the $\widehat G$ -conjugacy classes of relevant local L-parameters. The basic local Langlands correspondence predicts a surjective finite-to-one map

\operatorname{Irr}(G(F)) \longrightarrow \Phi(G), \qquad \pi\longmapsto\varphi_\pi,

from those representations to local $L$ -parameters. For a nonarchimedean $F$ , “admissible” means smooth admissible. The finite fiber

\Pi_\varphi(G)= \{\pi\in\operatorname{Irr}(G(F)):\varphi_\pi=\varphi\}

is the $L$ -packet of $\varphi$ .

Status

This is a theorem for archimedean reductive groups, tori, and $\operatorname{GL}_n$ , and it is established with refined structures for many classical groups and numerous other families. For a general reductive group over a nonarchimedean field, the full basic and refined statements remain partly conjectural.

General constructions due to Genestier–Lafforgue and Fargues–Scholze attach semisimplified parameters in broad settings. Questions such as surjectivity, packet finiteness, recovery of monodromy, and all expected character identities must still be distinguished from the existence of that semisimplified map.

Structural requirements

A correspondence is not characterized by cardinality alone. It is expected to respect unramified Satake parameters, central characters, twists, contragredients, temperedness, parabolic induction, local $L$ - and epsilon factors where defined, and endoscopic character identities. The refined correspondence supplies the internal structure of each packet.

Basic examples

For a torus, packets are singletons and the correspondence is local class field theory in dual-group form. For $\operatorname{GL}_n$ , packets are also singletons, but the parameter is an $n$ -dimensional Weil–Deligne representation.

References

Tasho Kaletha, “Representations of reductive groups over local fields,” §§2.1–2.3, 2022. arXiv.
Michael Harris, “On the local Langlands correspondence,” 2003. arXiv.
Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” 2021. arXiv.