Langlands reciprocity

The parametrization of tempered representations of $GL(n,{\mathbb R})$ can be reformulated as follows. We need to introduce the Weil group $W_{\mathbb R}$ of ${\mathbb R}$. This group is a non-split extension of the multiplicative group $\mathbb C^*$ of $\mathbb C$ by $Gal(\mathbb C/\mathbb R)=\mathbb Z/2\mathbb Z$. Explicitly, $W_{\mathbb R}={\mathbb C}^*\cup\sigma{\mathbb C}^*$, where $\sigma^2=-1$ and $\sigma z \sigma^{-1}=\overline{z}$ for $z\in\mathbb C^*$. An $n$ dimensional complex representation $\phi$ of $W_{\mathbb R}$ is said to be admissible if $\phi$ is continuous and if $\phi(W_{\mathbb R})$ consists of semisimple matrices. The local Langlands reciprocity conjecture for $GL(n,\mathbb R)$ (which is known to be true) states that the irreducible admissible representations of $GL(n,\mathbb R)$ are in 1-1 correspondence with the admissible n-dimensional representations of $W_{\mathbb R}$. This parametrization respects classes of representations with special properties. For instance, the tempered representations correspond to the unitary representations of $W_{\mathbb R}$, and the discrete series representations correspond to the representations of $W_{\mathbb R}$ whose image is not contained in any proper parabolic subgroup (i.e. a subgroup containing a conjugate of the subgroup of upper triangular matrices).

For general real reductive groups² $G(\mathbb R)$, Langlands has defined a dual reductive group ${}^LG$ (over $\mathbb C$) such that the conjugacy classes of admissible homomorphisms of the group $W_{\mathbb R}$ to ${}^LG$ should correspond to finite ``packets'' of ``L-indistinguishable'' irreducible admissible representations. What controls the size of these finite packets is not completely clear, and this complicates the precise formulation of the local Langlands correspondence in this generality considerably.

As a simple example, let us look at the case $GL(2,\mathbb R)$. Its dual group is $GL(2,\mathbb C)$. The unitary representations of $W_{\mathbb R}$ are easily seen to be conjugate to precisely one of the following

(1) $\sigma\rightarrow \pm\left(\begin{array}{cc} 1&0\\ 0&1\end{array}\right),\ z\... ... z\vert}^{i\lambda_2}\end{array}\right) \ (\lambda_1\leq\lambda_2\in\mathbb R)$
(2) $\sigma\rightarrow \left(\begin{array}{cc} 1&0\\ 0&-1\end{array}\right),\ z\ri... ...ert z\vert}^{i\lambda_2}\end{array}\right) \ (\lambda_1,\lambda_2\in\mathbb R)$
(3) $\sigma\rightarrow \left(\begin{array}{cc} 0&1\\ (-1)^l&0\end{array}\right),\ ... ...d{array}\right)\\ \hspace*{7mm} (l\in\mathbb Z_{\geq 1},\lambda\in\mathbb R).$

This list of possibilities corresponds nicely to the well known list of irreducible tempered representations of $GL(2,\mathbb R)$. The cases (1) and (2) correspond to spherical and non-spherical unitary principal series, and (3) corresponds to the discrete series.

The (slightly more complicated) classification of unitary representations of $SL(2,\mathbb R)$ was already known since 1947, by a famous paper of Bargmann. The results for $GL(2,\mathbb R)$ can be easily reconstructed from that paper. The fact that the irreducible tempered representations of $GL(2,\mathbb R)$ are in correspondence with a list of equivalence classes of continuous homomorphisms of $W_{\mathbb R}$ seems coincidental. After all, the construction of the various series of tempered representations and the proof of their completeness, is a delicate and complicated analytic task. The techniques and constructions that are employed have, at first sight, nothing in common with the set of equivalence classes of two dimensional unitary representations of $W_{\mathbb R}$.

Yet, in the 1960's Langlands has put forward his now famous reciprocity conjecture, asserting that

(1)

In all cases of a reductive form $G(F)$ over $F$, where $F$ is a local field, a ``local correspondence'' should exist, similar to what we have formulated for real reductive groups. We have to change the group $W_{\mathbb R}$ to the Weil group $W_F$ of $F$. This Weil group is a close relative of the Galois group $Gal(\overline{F}/F)$ of $F$. In the case of a p-adic field of characteristic $0$ for instance, the Weil group is a dense subgroup of the Galois group, but its topology is stronger than the relative topology (thus allowing more continuous representations).

(2)

There should even exist a ``global correspondence'' for $G(A_F)$, where $A_F$ is the adelic ring of a global field $F$. The admissible homomorphisms from the global Weil group $W_F$ to the Langlands dual group ${}^LG$ (up to conjugation in ${}^LG$) should now correspond to equivalence classes of so-called automorphic representations of $G(A_F)$.

In all cases, the definition of the dual group ${}^LG$ is the same.

Langlands reciprocity conjecture was inspired by the famous reciprocity law of Emil Artin. Artin's reciprocity law states precisely that the abelianized group $W_F^{ab}$ is isomorphic to $F^*=GL(1,F)$ in the local case, and to the idele class group $GL(1,F)\backslash GL(1,A_F)$ in the global case. We can restate this by saying that the unitary characters of $W_F$ should correspond bijectively to the unitary characters of $GL(1,F)$ (local case) or to the unitary characters of the idele class group (Hecke characters) (global case). It is the case $G=GL(1)$ of the Langlands reciprocity conjecture.

The conjecture has many refinements predicting how the correspondence behaves under restriction to subclasses of representations, base change, and most profoundly, homomorphisms between the dual groups ${}^LG$ (Langlands functoriality principle). Even locally this predicts, in a very precise way, properties of irreducible unitary representations.

Many special cases have been verified over the last three decades, but it is fair to say that it remains a mystery why it works.

Recently, the local Langlands correspondence for $GL(n,F)$ with $F$ a p-adic field was verified by Harris and Taylor, and Henniart. The correspondence was also verified for $GL(n)$ in the case of a global field of positive characteristic (Drinfeld, Lafforgue). Although these are certainly a very important steps forward, the methods that have been used are not applicable to the general reductive case.