# 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