The Local Langlands Correspondence for General Linear Groups

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 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 “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.

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

Langlands Correspondence and Bezrukavnikov Equivalence by Anna Romanov and Geordie Williamson

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

Representation Theory and Fourier Analysis

In Some Basics of Fourier Analysis we introduced some of the basic ideas in Fourier analysis, which is ubiquitous in many parts of both pure and applied math. In this post we look at these same ideas from a different point of view, that of representation theory.

Representation theory is a way of studying group theory by turning it into linear algebra, which in many cases is more familiar to us and easier to study.

A (linear) representation is just a group homomorphism from some group $G$ we’re interested in, to the group of linear transformations of some vector space. If the vector space has some finite dimension $n$, the group of its linear transformations can be expressed as the group of $n \times n$ matrices with nonzero determinant, also known as $\mathrm{GL}_n(k)$ ($k$ here is the field of scalars of our vector space).

In this post, we will focus on infinite-dimensional representation theory. In other words, we will be looking at homomorphisms of a group $G$ to the group of linear transformations of an infinite-dimensional vector space.

“Infinite-dimensional vector spaces” shouldn’t scare us – in fact many of us encounter them in basic math. Functions are examples of such. After all, vectors are merely things we can scale and add to form linear combinations. Functions satisfy that too. That being said, if we are dealing with infinity we will often need to make use of the tools of analysis. Hence functional analysis is often referred to as “infinite-dimensional linear algebra” (see also Metric, Norm, and Inner Product).

Just as a vector $v$ has components $v_i$ indexed by $i$, a function $f$ has values $f(x)$ indexed by $x$. If we are working over uncountable things, instead of summation we may use integration.

We will also focus on unitary representations in this post. This means that the linear transformations are further required to preserve a complex inner product (which takes the form of an integral) on the vector space. To facilitate this, our functions must be square-integrable.

Consider the group of real numbers $\mathbb{R}$ (under addition). We want to use representation theory to study this group. For our purposes we want the square-integrable functions on some quotient of $\mathbb{R}$ as our vector space. It comes with an action of $\mathbb{R}$, by translation. In other words, an element $a$ of $\mathbb{R}$ acts on our function $f(x)$ by sending it to the new function $f(x+a)$.

So what is this quotient of $\mathbb{R}$ that our functions will live on? For now let us choose the integers $\mathbb{Z}$. The quotient $\mathbb{R}/\mathbb{Z}$ is the circle, and functions on it are periodic functions.

To recap: We have a representation of the group $\mathbb{R}$ (the real line under addition) as linear transformations (also called linear operators) of the vector space of square-integrable functions on the circle.

In representation theory, we will often decompose a representation into a direct sum of irreducible representations. Irreducible means it contains no “subrepresentation” on a smaller vector space. The irreducible representations are the “building blocks” of other representations, so it is quite helpful to study them.

How do we decompose our representation into irreducible representations? Consider the representation of $\mathbb{R}$ on the vector space $\mathbb{C}$ (the complex numbers) where a real number $a$ acts by multiplying a complex number $z$ by $e^{2\pi i k a}$, for $k$ an integer. This representation is irreducible.

If this looks familiar, this is just the Fourier series expansion for a periodic function. So a Fourier series expansion is just an expression of the decomposition of the representation of R into irreducible representations!

What if we chose a different vector space instead? It might have been the more straightforward choice to represent $\mathbb{R}$ via functions on $\mathbb{R}$ itself instead of on the circle $\mathbb{R}/\mathbb{Z}$. That may be true, but in this case our decomposition into irreducibles is not countable! The irreducible representations into which this other representation decomposes is the one where a real number $a$ acts on $\mathbb{C}$ by multiplication by $e^{2 \pi i k a}$ where $k$ is now a real number, not necessarily an integer. So it’s not indexed by a countable set.

This should also look familiar to those who know Fourier analysis: This is the Fourier transform of a square-integrable function on $\mathbb{R}$.

So now we can see that concepts in Fourier analysis can also be phrased in terms of representations. Important theorems like the Plancherel theorem, for example, also may be understood as an isomorphism between the representations we gave and other representations on functions of the indices. We also have the Poisson summation in Fourier analysis. In representation theory this is an equality obtained from calculating the trace in two ways, as a sum over representations and as a sum over conjugacy classes.

Now we see how Fourier analysis is related to the infinite-dimensional representation theory of the group $\mathbb{R}$ (one can also see this as the infinite-dimensional representation theory of the circle, i.e. the group $\mathbb{R}/\mathbb{Z}$ – the article “Harmonic Analysis and Group Representations” by James Arthur discusses this point of view). What if we consider other groups instead, like, say, $\mathrm{GL}_n(\mathbb{R})$ or $\mathrm{SL}_n(\mathbb{R})$ (or $\mathbb{R}$ can be replaced by other rings even)?

Things get more complicated, for example the group may not be abelian. Since we used integration so much, we also need an analogue for it. So we need to know much about group theory and analysis and everything in between for this.

These questions have been much explored for the kinds of groups called “reductive”, which are closely related to Lie groups. They include the examples of $\mathrm{GL}_n(\mathbb{R})$ and $\mathrm{SL}_n(\mathbb{R})$ earlier, as well as certain other groups we have discussed in previous posts such as the orthogonal and unitary (see also Rotations in Three Dimensions). There is a theory for these groups analogous to what I have discussed in this post, and hopefully this will be discussed more in future blog posts here.

References:

Representation theory on Wikipedia

Representation of a Lie group on Wikipedia

Fourier analysis on Wikipedia

Harmonic analysis on Wikipedia

Plancherel theorem on Wikipedia

Poisson summation formula on Wikipedia

An Introduction to the Trace Formula by James Arthur

Harmonic Analysis and Group Representations by James Arthur