Global Langlands functoriality conjecture

Let $F$ be a number field with adele ring $\mathbb A_F$ and absolute Galois group $\Gamma_F=\operatorname{Gal}(\bar F/F)$. For a connected reductive group $G$ over $F$, let $\hat G$ be its complex dual group, with the action of $\Gamma_F$ obtained from a pinning. The $L$-group is the semidirect product

$${}^LG=\hat G\rtimes\Gamma_F .$$

Let $H$ and $G$ be quasi-split connected reductive groups over $F$. An $L$-homomorphism $\rho\colon{}^LH\to{}^LG$ is a continuous homomorphism that commutes with the projections to $\Gamma_F$ and whose restriction to $\hat H$ is a morphism of complex algebraic groups.

An automorphic representation of $G(\mathbb A_F)$ is an irreducible constituent of the space of automorphic forms on $G(F)\backslash G(\mathbb A_F)$, in the sense of Borel and Jacquet [3]. It factors as $\pi=\bigotimes_v'\pi_v$. Let $S$ be a finite set of places containing the archimedean places and the places where $G$, $H$ or the given representations ramify. For $v\notin S$, $\pi_v$ is unramified, and the Satake isomorphism attaches to it a semisimple $\hat G$-conjugacy class

$$c(\pi_v)\subset\hat G\rtimes\operatorname{Frob}_v\subset{}^LG .$$

Conjecture (global functoriality). For every $F$, $H$, $G$ and $\rho$ as above, and every automorphic representation $\pi'$ of $H(\mathbb A_F)$, there is an automorphic representation $\pi$ of $G(\mathbb A_F)$ and a finite set of places $S$ such that

$$c(\pi_v)=\rho\bigl(c(\pi'_v)\bigr)\qquad\text{for every place } v\notin S .$$

This is the weak, almost-everywhere form [1, 4]. It implies the equality of partial $L$-functions

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

for every finite-dimensional representation $r$ of ${}^LG$. A stronger form asks that $\pi_v$ correspond to $\pi'_v$ under local functoriality at every place. The problem here is the conjecture as stated, for all $H$, $G$ and $\rho$. Many individual cases are theorems, listed below.

Background

Langlands formulated functoriality in his 1967 letter to Weil and in his 1970 paper "Problems in the theory of automorphic forms" [1]. Borel's Corvallis article fixed the language of $L$-groups and $L$-homomorphisms used today [2]. Arthur's surveys explain the principle and its place in the Langlands program [4, 14].

The case $H=\{1\}$ contains Artin's conjecture. A continuous representation $\sigma\colon\Gamma_F\to\mathrm{GL}_n(\mathbb C)$ with finite image defines an $L$-homomorphism ${}^L\{1\}=\Gamma_F\to{}^L\mathrm{GL}_n$. Functoriality then gives an automorphic representation of $\mathrm{GL}_n(\mathbb A_F)$ with the same $L$-function. This implies Artin's conjecture that $L(s,\sigma)$ is entire when $\sigma$ is irreducible and nontrivial. Functoriality for the tensor product map $\mathrm{GL}_m\times\mathrm{GL}_n\to\mathrm{GL}_{mn}$ and for symmetric powers would imply the generalized Ramanujan conjecture for $\mathrm{GL}_n$.

Most established cases use one of three tools: the trace formula and its stabilization, converse theorems combined with Langlands–Shahidi $L$-functions, or modularity lifting for Galois representations.