May | 2021 | Theories and Theorems

The Langlands program is, roughly, a part of math that bridges the representation theory of reductive groups and Galois representations. It consists of many interrelated conjectures, among which are the global and local Langlands correspondences, which may be seen as higher-dimensional analogues of global and local class field theory (see also Some Basics of Class Field Theory) in some way. Just as in class field theory, the words “global” and “local” refer to the fact that these correspondences involve global and local fields, respectively, and their absolute Galois groups.

In this post we will discuss the local correspondence (the global correspondence requires the notion of automorphic forms, which involves a considerable amount of setup, hence is postponed to future posts). We will also only discuss the case of the general linear group $\mathrm{GL}_{n}(F)$ , for $F$ a local field, and in particular $\mathrm{GL}_{1}(F)$ and $\mathrm{GL}_{2}(F)$ where we can be more concrete. More general reductive groups will bring in the more complicated notions of L-packets and L-parameters, and therefore also postponed to future posts.

Let us give the statement first, and then we shall unravel what the words in the statement mean.

The local Langlands correspondence for general linear groups states that there is a canonical one-to-one correspondence between irreducible admissible representations of $\mathrm{GL}_{n}(F)$ (over $\mathbb{C}$ ) and F-semisimple Weil-Deligne representations of the Weil group $W_{F}$ (also over $\mathbb{C}$ ).

Let us start with “canonical”. This means that our one-to-one correspondence isn’t just any one-to-one correspondence, but relates certain invariants of either side (usually expressed in terms of L-functions). We will leave the details of these invariants to future posts, but we will give an explicit example of this bijection later.

Next let us explain “irreducible admissible representations of $\mathrm{GL}_{n}(F)$ “. These representations are similar to what we discussed in Representation Theory and Fourier Analysis (in fact as we shall see, many of these representations are also infinite-dimensional, and constructed somewhat similarly). Just to recall, irreducible means that the only subspaces held fixed by $\mathrm{GL}_{n}(F)$ are $0$ and the entire subspace.

Admissible means that, if we equip $\mathrm{GL}_{n}(F)$ with the topology that comes from the $p$ -adic topology of the field $F$ , for any open $U$ subgroup of $\mathrm{GL}_{n}(F)$ the fixed vectors form a finite-dimensional subspace.

Closely related to the notion of admissible is the notion of smooth. A representation of $\mathrm{GL}_{n}(F)$ over some vector space $V$ is smooth if for any $v\in V$ the stabilizer of $v$ is an open subset of $\mathrm{GL}_{n}(F)$ . It is known that, in the case we are considering, an irreducible smooth representation is automatically irreducible.

Now we look at the other side of the correspondence. We already defined what a Weil-Deligne representation is in Weil-Deligne Representations. A Weil-Deligne representation $(\rho_{0},N)$ is F-semisimple if the representation $\rho_{0}$ is the direct sum of irreducible representations.

In the case of $\mathrm{GL}_{1}(F)$ , the local Langlands correspondence is a restatement of local class field theory. We have that $\mathrm{GL}_1(F)=F^{\times}$ , and the only irreducible admissible representations of $\mathrm{GL}_1(F)$ are continuous group homomorphisms $\chi:F^\times\to\mathbb{C}^{\times}$ .

On the other side of the correspondence we have the one-dimensional Weil-Deligne representations $(\rho_{0},N)$ of $W_F$ , which must have monodromy operator $N=0$ and must factor through the abelianization $W_F^{\mathrm{ab}}$ .

Recall from Weil-Deligne Representations that we local class field theory gives us an isomorphism $\mathrm{rec}:F^{\times}\xrightarrow{\sim}W_{F}^{\mathrm{ab}}$ , also known as the Artin reciprocity map. We can now describe the local Langlands correspondence explicitly. It sends $\chi$ to the Weil-Deligne representation $(\rho_{0},0)$ , where $\rho_{0}$ is the composition $W_{F}\to W_{F}^{\mathrm{ab}}\xrightarrow{\mathrm{rec}^{-1}}F^{\times}\xrightarrow{\chi}\mathbb{C}^{\times}$ .

Now let us consider the case of $\mathrm{GL}_{2}(F)$ . If the residue field of $F$ is not of characteristic $2$ , then the irreducible admissible representations of $\mathrm{GL}_{2}(F)$ may be enumerated, and they fall into four types: principal series, special, one-dimensional, and supercuspidal.

Let $\chi_{1},\chi_{2}:F^{\times}\to\mathbb{C}^{\times}$ be continuous admissible characters and let $I(\chi_{1},\chi_{2})$ be the vector space of functions $\phi:\mathrm{GL}_{2}(F)\to\mathbb{C}$ such that

$\displaystyle \phi \left(\begin{pmatrix}a&b\\0&d\end{pmatrix}g\right)=\chi_{1}(a)\chi_{2}(d)\Vert a/d\Vert^{1/2}\phi(g)$

The group $\mathrm{GL}_{2}(F)$ acts on the functions $\phi$ , just as in Representation Theory and Fourier Analysis. Therefore it gives us a representation of $\mathrm{GL}_{2}(F)$ on the vector space $I(\chi_{1},\chi_{2})$ , which we say is in the principal series.

Now the representation $I(\chi_{1},\chi_{2})$ might be irreducible, in which case it is one of the things that go into our correspondence, or it might be reducible. This is decided by the Bernstein-Zelevinsky theorem, which says that $I(\chi_{1},\chi_{2})$ is irreducible precisely if the ration of the characters $\chi_{1}$ and $\chi_{2}$ is not equal to plus or minus $1$ .

In the case that $\chi_{1}/\chi_{2}=1$ , then we have an exact sequence

$\displaystyle 0\to\rho\to I(\chi_{1},\chi_{2})\to S(\chi_{1},\chi_{2})\to 0$

where the representations $S(\chi_{1},\chi_{2})$ and $\rho$ are both irreducible representations of $\mathrm{GL}_{2}(F)$ . The representation $S(\chi_{1},\chi_{2})$ is infinite-dimensional and is known as the special representation. The representation $\rho$ is the one-dimensional representation and is given by $\chi_{1}\Vert\cdot\Vert^{1/2}\det$ .

If $\chi_{1}/\chi_{2}=-1$ instead, then we have a “dual” exact sequence

$\displaystyle 0\to S(\chi_{1},\chi_{2}) \to I(\chi_{1},\chi_{2})\to \rho\to 0$

So far the irreducible admissible representations of $\mathrm{GL}_{2}(F)$ that we have seen all arise as subquotients of $I(\chi_{1},\chi_{2})$ . Since characters such as $\chi_{1}$ and $\chi_{2}$ are the irreducible admissible representations of $\mathrm{GL}_{1}(F)$ , we may consider the principal series, special, and one-dimensional representations as being built out of these more basic building blocks.

However there exist irreducible admissible representations that do not arise via this process, and they are called supercuspidal representations. For $\mathrm{GL}_{2}(F)$ there is one kind of supercuspidal representation denoted $\mathrm{BC}_{E}^{F}(\psi)$ for $E$ a quadratic extension of $F$ and $\psi$ an admissible character $\psi:E\to\mathbb{C}^{\times}$ .

Now we know what the irreducible admissible representations of $\mathrm{GL}_{2}(F)$ are. The local Langlands correspondence says that they will correspond to F-semisimple Weil-Deligne representations. We can actually describe explicitly which Weil-Deligne representation each irreducible admissible representation of $\mathrm{GL}_{2}(F)$ gets sent to!

Let $\chi_{1},\chi_{2}:F^{\times}\to \mathbb{C}^{\times}$ be the same continuous admissible characters used to construct the irreducible representations as above, and let $\rho_{1},\rho_{2} :W_{F}\to \mathbb{C}^{\times}$ be the corresponding representation of the Weil group given by the local Langlands correspondence for $\mathrm{GL}_{1}$ , as discussed earlier. Then to each irreducible admissible representation of $\mathrm{GL}_2(F)$ we associate a $2$ -dimensional Weil-Deligne representation as follows:

To the principal series representation $I(\chi_{1},\chi_{2})$ we associate the Weil-Deligne representation $(\rho_{1}\oplus\rho_{2},0)$ .

To the special representation $S(\chi_{1},\chi_{1}\times\Vert\cdot\Vert)$ , we associate the Weil-Deligne representation $\left(\begin{pmatrix}\Vert\cdot\Vert\rho_{1} & 0\\0 & \rho_{1}\end{pmatrix},\begin{pmatrix} 0 & 1\\0 & 0\end{pmatrix}\right)$ .

To the one-dimensional representation $\chi_{1}\circ\det$ , we associate the Weil-Deligne representation $\left(\begin{pmatrix}\rho_{1}\times\Vert\cdot\Vert^{1/2} & 0\\0 & \rho_{1}\times\Vert\cdot\Vert^{-1/2}\end{pmatrix},0\right)$ .

Finally, to the supercuspidal representation $\mathrm{BC}_{E}^{F}(\psi)$ we associate the Weil-Deligne representation $(\mathrm{Ind}_{W_{E}}^{W_{F}}\sigma,0$ ), where $\sigma$ is the unique nontrivial element of $\mathrm{Gal}(E/F)$ .

We have been able to describe the local Langlands correspondence for $\mathrm{GL}_{1}(F)$ and $\mathrm{GL}_{2}(F)$ explicitly (in the latter case as long as the characteristic of the residue field of $F$ is not $2$ ). The local Langlands correspondence for $\mathrm{GL}_{n}(F)$ , for more general $n$ on the other hand, was proven via geometric means – namely using the geometry of certain Shimura varieties (see also Shimura Varieties) as well as their local counterpart, the Lubin-Tate tower, which parametrizes deformations of Lubin-Tate formal group laws (see also The Lubin-Tate Formal Group Law) together with level structure.

There has been much recent work regarding the local Langlands program for groups other than $\mathrm{GL}_{n}(F)$ . For instance there is work on the local Langlands correspondence for certain symplectic groups making use of “theta lifts”, by Wee Teck Gan and Shuichiro Takeda. Very recently, there has also been work by Laurent Fargues and Peter Scholze that makes use of ideas from the geometric Langlands program. These, and more, will hopefully be discussed more here in the future.

References:

Langlands program on Wikipedia

Local Langlands conjectures on Wikipedia

Local Langlands conjecture on the nLab

MSRI Summer School on Automorphic Forms and the Langlands Program by Kevin Buzzard

Langlands Correspondence and Bezrukavnikov Equivalence by Anna Romanov and Geordie Williamson

The Local Langlands Conjecture for GL(2) by Colin Bushnell and Guy Henniart

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Month: May 2021

The Local Langlands Correspondence for General Linear Groups