Math Conjectures

Global Langlands Functoriality Conjecture

OPENLandmarkConjectureProposed 1967–1970 · Standard version

Canonical statement

Let FF be a global field, let GG and HH be connected reductive groups over FF, with HH quasi-split, and let r:LG→LHr:{}^LG\to{}^LH be an admissible LL-homomorphism compatible with the projections to the global Weil group. For every automorphic representation π=⨂vπv\pi=\bigotimes_v\pi_v of G(AF)G(\mathbb A_F), there exists an automorphic representation Π=⨂vΠv\Pi=\bigotimes_v\Pi_v of H(AF)H(\mathbb A_F) such that, at every place vv where all data are unramified, their Satake conjugacy classes satisfy
s(Πv)=r(s(πv)). s(\Pi_v)=r\bigl(s(\pi_v)\bigr).
View source LaTeX
Let \(F\) be a global field, let \(G\) and \(H\) be connected reductive groups over \(F\), with \(H\) quasi-split, and let \(r:{}^LG\to{}^LH\) be an admissible \(L\)-homomorphism compatible with the projections to the global Weil group. For every automorphic representation \(\pi=\bigotimes_v\pi_v\) of \(G(\mathbb A_F)\), there exists an automorphic representation \(\Pi=\bigotimes_v\Pi_v\) of \(H(\mathbb A_F)\) such that, at every place \(v\) where all data are unramified, their Satake conjugacy classes satisfy
\[
  s(\Pi_v)=r\bigl(s(\pi_v)\bigr).
\]
NT-AUTO-038:[Langlands1970AutomorphicProblems][Arthur2003Functoriality][Ngo2020HankelFunctoriality]
Number theoryAlgebra and representation theory

Notes

Langlands functoriality predicts that an admissible map r:LG→LHr:{}^LG\to{}^LH between LL-groups transports automorphic representations of GG to automorphic representations of HH. At an unramified place, an automorphic representation is encoded by its Satake conjugacy class, so the canonical weak requirement is the exact relation s(Πv)=r(s(πv))s(\Pi_v)=r(s(\pi_v)). This transfer principle emerged from Langlands's foundational formulation of automorphic reciprocity [Langlands1970AutomorphicProblems].

Functoriality is known in important but structured settings: cyclic base change, endoscopic transfer, and many transfers involving classical groups or general linear groups. Arthur explains how these cases fit the general principle [Arthur2003Functoriality], while later work develops analytic mechanisms for functorial transfer and associated functional equations [Ngo2020HankelFunctoriality].

No construction handles every global field, connected reductive pair, and admissible LL-homomorphism. The entry deliberately fixes the weak unramified statement; assertions about ramified local parameters, packets, multiplicities, cuspidality, and full local–global compatibility are refinements of the wider Langlands program and are not silently included.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.