Langlands functoriality and L-homomorphisms

Let $G'$ and $G$ be connected reductive groups over a local or global field $F$ . An $L$ -homomorphism between their $L$ -groups is a map

\omega:{}^LG'\longrightarrow{}^LG

$\omega$ commutes with the projections to the Weil or Galois factor and is algebraic on the dual-group identity components. It is considered up to $\widehat G$ -conjugacy and may require the standard admissibility conditions.

Parameter-level statement

The most direct operation is composition:

\varphi'\longmapsto\omega\circ\varphi'.

Locally, functoriality predicts that the $L$ -packet for the local $L$ -parameter $\varphi'$ transfers to representations in the packet for $\omega\circ\varphi'$ . For groups with nontrivial packets this is not naturally a function from one individual representation to one individual representation without additional data.

Global transfer

Globally, if $\pi'$ is an automorphic representation of $G'(\mathbb A_F)$ , functoriality predicts an automorphic representation or packet on $G(\mathbb A_F)$ whose unramified Satake parameters satisfy

c(\pi_v)\sim\omega(c(\pi_v'))

at almost every place. A complete transfer should also have the expected ramified and archimedean local behavior.

L-functions

For $r:{}^LG\to\operatorname{GL}(V)$ , the identity between incomplete Euler products

L^S(s,\pi,r) = L^S(s,\pi',r\circ\omega)

follows formally from matching almost-all unramified parameters. Equality of incomplete $L$ -functions alone does not establish the full local or packet-level transfer.

Status and methods

General functoriality remains conjectural. Established families arise from cyclic base change and automorphic induction, endoscopic classification, theta correspondences, converse theorems, trace-formula comparisons, and specific symmetric-power or tensor-product constructions. Endoscopy is a structured part of functoriality, not the whole principle.

Relation to the letter

This is the letter's second broad question. Its unramified formulation already contains the modern parameter-level idea; the later theory adds local packets, stable trace formulas, Arthur parameters, and explicit theorem-status boundaries.

References

Robert P. Langlands, “Problems in the theory of automorphic forms,”
IAS copy.
James Arthur, “The principle of functoriality,” 2002. Clay copy.