# Reductive Groups Part II: Over More General Fields

In Reductive Groups Part I: Over Algebraically Closed Fields we learned about how reductive groups over algebraically closed fields are classified by their root datum, and how the based root datum helps us understand their automorphisms. In this post, we consider reductive groups over more general, not necessarily algebraically closed fields. We will discuss how they can be classified, and also define the Langlands dual of a reductive group, which will allow us to state the local Langlands correspondence (see The Local Langlands Correspondence for General Linear Groups) for certain groups other than just the general linear group.

Let $F$ be a field. We will say an algebraic group $G$ over $F$ is a reductive group if $G_{\overline{F}}$, the base change of $G$ to the algebraic closure $\overline{F}$, is a reductive group. Similarly, we say that $G$ is a torus if the base change $G_{\overline{F}}$ is a torus. This means that after base change to the algebraic closure it becomes isomorphic to the product of copies of the multiplicative group $\mathbb{G}_{m}$. However, if over $F$ it is already isomorphic to the product of copies of $\mathbb{G}_{m}$, without the need for a base change, then we say that it is a split torus. If a reductive group $G$ contains a maximal split torus, we say that $G$ is split. We note that a “maximal split torus” is different from a “split maximal torus”!

The classification of split reductive groups is the same as that of reductive groups over algebraically closed fields – they are classified by their root datum. As such they will provide us with the first step towards classifying reductive groups over more general fields.

If $G$ is a reductive group over $F$, a form of $G$ is some other reductive group $G'$ over $F$ such that after base change to the algebraic closure $\overline{F}$, $G_{\overline{F}}$ and $G_{\overline{F}}$ are isomorphic. It happens that any reductive group is a form of a split reductive group. This follows from the fact that any abstract root datum is the root datum associated to some reductive group and some split maximal torus contained in it.

The forms of a split group are classified using Galois cohomology. Suppose we have an isomorphism $f:G_{\overline{F}}\simeq G_{\overline{F}}$. The Galois group $\mathrm{Gal}(\overline{F}/F)$ (henceforth shortened to just $\mathrm{Gal}_{F})$ acts on the isomorphism $f$ by conjugation, giving rise to another isomorphism $^{\sigma}f:G_{\overline{F}}\simeq G_{\overline{F}}'$. Composing this with the inverse of $f$ we get an automorphism $f^{-1}\circ^{\sigma}f$ of $G_{\overline{F}}$. This automorphism is an example of a $1$cocycle in Galois cohomology.

More generally, in Galois cohomology, for some group $M$ with a Galois action (for instance in our case $M=\mathrm{Aut}(G)_{\overline{F}})$), a $1$-cocycle is a homomorphism $\varphi:\mathrm{Gal}_{F}\to M$ such that $\varphi(\sigma\tau)=\varphi(\sigma)\cdot^{\sigma}\varphi(\tau)$. Two $1$-cocycles $\varphi, \psi$ are cohomologous if there is an element $m\in M$ such that $\psi(\sigma)=m^{-1}\varphi(\sigma)^{\sigma}m$. The set of $1$-cocycles, modulo those which are cohomologous, is denoted $H^{1}(\mathrm{Gal}_{F},M)$.

By the above construction there is a map between the set of isomorphism classes of forms of $G$ and the Galois cohomology group $H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}}))$. This map actually happens to be a bijection!

Let $BR$ be a based root datum corresponding to $G$ together with a pinning. We have mentioned in Reductive Groups Part I: Over Algebraically Closed Fields the group of automorphisms of $BR$, the pinned automorphisms of $G$, and the outer automorphisms of $G$ are all isomorphic to each other. We have the following exact sequence

$\displaystyle 0\to\mathrm{Inn}(G_{\overline{F}})\to\mathrm{Aut}(G_{\overline{F}})\to\mathrm{Out}(G_{\overline{F}})\to 0$

and when we are provided the additional data of a pinning this gives us a splitting of the exact sequence (i.e. a way to decompose the middle term into a semidirect product of the other two terms).

When the pinning is defined over $F$, we obtain a homomorphism

$\displaystyle H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}}))\to H^{1}( \mathrm{Gal}_{F}, \mathrm{Out}(G_{\overline{F}}))$

where $H^{1}(\mathrm{Gal}_{F}, \mathrm{Out}(G_{\overline{F}}))$ is in bijection with the set of conjugacy classes of group homomorphisms $\mu:\mathrm{Gal}_{F}\to \mathrm{Out}(G_{\overline{F}})$. But we have said earlier that $H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}}))$ is in bijection with the set of isomorphism classes of forms of $G$. Therefore, any form of $G$ gives us such a homomorphism $\mu:\mathrm{Gal}_{F}\to \mathrm{Out}(G_{\overline{F}})$.

We say that a reductive group is quasi-split if it contains a Borel subgroup. Split reductive groups are automatically quasi-split.

An inner form of a reductive group $G$ is another reductive group $G'$ related by an isomorphism $f:G_{\overline{F}}\simeq G_{\overline{F}}'$ such that the composition $f^{-1}\circ^{\sigma}f$ is an inner automorphism of $G_{\overline{F}}$.

Once we have a split group $G$, and given the data of a pinning, we can now use any morphism $\mu:\mathrm{Gal}_{F}\to\mathrm{Out}(G)$ together with the given pinning to obtain an element of $H^{1}(\mathrm{Gal}_{F},\mathrm{Aut}(G_{\overline{F}}))$, which in turn will give us a quasi-split form of $G$. Now it happens that any reductive group $G$ has a unique quasi-split inner form!

Therefore, in summary, the classification of reductive groups over general fields proceeds in the following three steps:

1. Classify the split reductive groups using the root datum.
2. Classify the quasi-split forms using the homomorphisms $\mathrm{Gal}_{F}\to\mathrm{Out}(G)$.
3. Classify the inner forms of the quasi-split forms.

Let us now discuss the Langlands dual (also known as the L-group) of a reductive group. Since every abstract root datum corresponds to some reductive group $G$ (say, over a field $F$), we can interchange the roots and coroots and get another reductive group $\widehat{G}$, which we refer to as the dual group of $G$.

The Langlands dual of $G$ is the group (an honest to goodness group, not an algebraic group) given by the semidirect product $\widehat{G}(\mathbb{C})\rtimes \mathrm{Gal}_{F}$. In order to construct this semidirect product we need an action of $\mathrm{Gal}_{F}$ on $\widehat{G}(\mathbb{C})$, and in this case this action is via its action on the based root datum of $\widehat{G}$ together with a Borel subgroup $B\subseteq G$, which is the same as a pinned automorphism of $\widehat{G}$. We denote the Langlands dual of $G$ by $^{L}G$.

Let us recall that in The Local Langlands Correspondence for General Linear Groups we stated the local Langlands correspondence, in the case of $\mathrm{GL}_{n}(F)$ where $F$ is a local field, as a correspondence between the irreducible admissible representations of $\mathrm{GL}_{n}(F)$ (over $\mathbb{C}$) and the F-semisimple Weil-Deligne representations of the Weil group $W_{F}$ of $F$.

With the definition of the Langlands dual in hand, we can now state the local Langlands correspondence more generally, not just for $\mathrm{GL}_{n}(F)$, and in this case it will not even be a one-to-one correspondence between irreducible admissible representations and Weil-Deligne representations anymore!

First, we will need the notion of a Langlands parameter, also called an L-parameter, which takes the place of the F-semisimple Weil-Deligne representation. It is defined to be a continuous homomorphism $W_{F}\times \mathrm{SL}_{2}(\mathbb{C})\to ^{L}G$ such that, as a homomorphism from $W_{F}$ to $^{L}G$, it is semisimple, the composition $W_{F}\to^{L}G\to\mathrm{Gal}_{F}$ is just the usual inclusion of $W_{F}$ into $\mathrm{Gal}_{F}$, and as a function of $\mathrm{SL}_{2}(\mathbb{C})$ to $\widehat{G}(\mathbb{C})$ it comes from a morphism of algebraic groups from $\mathrm{SL}_{2}$ to $\widehat{G}$.

And now for the statement: The local Langlands correspondence states that, for a reductive group $G$ over a local field $F$, the irreducible admissible representations of $G(F)$ are partitioned into a finite disjoint union of sets, called L-packets, labeled by (equivalence classes of, where the equivalence is given by conjugation by elements of $\widehat{G}(\mathbb{C})$) L-parameters. In other words, letting $\mathrm{Irr}_{G}$ be the set of isomorphism classes of irreducible admissible representations of $G$, and letting $\Phi$ be the set of equivalence classes of L-parameters, we have

$\mathrm{Irr}_{G}=\coprod_{\phi\in\Phi}\Pi_{\phi}$

where $\Pi_{\phi}$ is the L-packet, a set of irreducible admissible representations of $G(F)$. In the case that $F$ is p-adic and $G=\mathrm{GL}_{n}$, each of these L-packets have only one element and this reduces to the one-to-one correspondence which we saw in The Local Langlands Correspondence for General Linear Groups.

The L-group and L-parameters are also expected to play a part in the global Langlands correspondence (in the case of function fields over a finite field, the construction of L-parameters was developed by Vincent Lafforgue using excursion operators). There is also much fascinating theory connecting the representations of the L-group to the geometry of a certain geometric object constructed from the original reductive group called the affine Grassmannian. We will discuss more of these topics in the future.

References:

Reductive group on Wikipedia

Root datum on Wikipedia

Inner form on Wikipedia

Langlands dual group on Wikipedia

Topics in automorphic forms (notes by Chao Li from a course by Jack Thorne)

Representation theory and number theory (notes by Chao Li from a course by Benedict Gross)

Algebraic groups by J. S. Milne

# 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