# 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

# Reductive Groups Part I: Over Algebraically Closed Fields

In many posts on this blog we have talked about elliptic curves, which are examples of algebraic groups (which is itself a specific instance of a group scheme) – a variety whose points form a group. As varieties, elliptic curves (and abelian varieties in general) are projective. There are also algebraic groups which are affine, and in this post we will discuss a special class of affine algebraic groups, called reductive groups, which as we shall see are related to many familiar objects, and are well-studied. In particular, in the case when they are defined over algebraically closed fields we will discuss their classification in terms of root datum. We will also discuss how this root datum helps us understand the automorphisms of such a reductive group.

An example of a reductive group is $\mathrm{GL}_{n}$; let’s assume that this is a variety defined over some field $F$. If $R$ is some $F$-algebra, then the $R$-valued points of $\mathrm{GL}_{n}$ (in the “functor of points” point of view) is the group $\mathrm{GL}_{n}(R)$ of $n\times n$ matrices with nonzero determinant. Geometrically, we may think of the nonzero determinant condition as the polynomial equation that cuts out the variety $\mathrm{GL}_{n}$.

Linear algebraic groups are smooth closed algebraic subgroups of $\mathrm{GL}_{n}$, and they have their own “representation theory”, a “representation” in this context being a morphism from some linear algebraic group $G$ to the algebraic group $\mathrm{GL}(V)$, for some vector space $V$ over some field $E$. The algebraic group $\mathrm{GL}(V)$ is the algebraic group whose $R$-valued points give the group of linear transformations of the $E$-vector space $R\otimes V$.

A linear algebraic group is a reductive group if it is geometrically connected and every representation is semisimple (a direct product of irreducible representations).

We also denote the reductive group $\mathrm{GL}_{1}$ by $\mathbb{G}_{m}$. A torus is a reductive group which is isomorphic to a product of copies of $\mathbb{G}_m$. A torus contained in a reductive group $G$ is called maximal if it is not contained in some strictly larger torus contained in $G$.

Let $T$ be a maximal torus of the reductive group $G$. The Weyl group $(G,T)$ is the quotient $N(T)/Z(T)$ where $N(T)$ is the normalizer of $T$ in $G$ (the subgroup consisting of all elements $g$ in $G$ such that for any element $t$ in $T$ $gtg^{-1}$ is an element of $T$) and $Z(T)$ is the centralizer of $T$ in $G$ (the subgroup consisting of all elements in $G$ that commute with all the elements of $T$).

Now let us discuss the classification of reductive groups, for which we will need the concept of roots and root datum.

For a maximal torus $T$ in a reductive group $G$, the characters (homomorphisms from $T$ to $\mathbb{G}_{m}$) and the cocharacters (homomorphisms from $T$ to $\mathbb{G}_{m}$) will play an important role in this classification. Let us denote the characters of $T$ by $X^{*}(T)$, and the cocharacters of $T$ by $X_{*}(T)$.

Just like Lie groups, reductive groups have a Lie algebra (the tangent space to the identity), on which it acts (therefore giving a representation of the reductive group, called the adjoint representation). We may restrict to a maximal torus $T$ contained in the reductive group $G$, so that the Lie algebra $\mathfrak{g}$ of $G$ gives a representation of $T$. This gives us a decomposition of $\mathfrak{g}$ as follows:

$\displaystyle \mathfrak{g}=\mathfrak{g}_{0}\oplus \bigoplus_{\alpha}\mathfrak{g}_{\alpha}$

Here $\mathfrak{g}_{\alpha}$ is the subspace of $\mathfrak{g}$ on which $T$ acts as a character $\alpha:T\to\mathbb{G}_{m}$. The nonzero characters $\alpha$ for which $\mathfrak{g}_{\alpha}$ is nonzero are called roots. We denote the set of roots by $\Phi$.

For a character $\alpha$, let $T_{\alpha}$ be the connected component of the kernel of $\alpha$. Let $G_{\alpha}$ be the centralizer of $T_{\alpha}$ in $G$. Then the Weyl group $W(G_{\alpha},T)$ will only have two elements, the identity and one other element, which we shall denote by $s_{\alpha}$. There will be a unique cocharacter $\alpha^{\vee}$ satisfying the equation

$s_{\alpha}(x)=x-\langle \alpha^{\vee},x\rangle\alpha$

for all characters $x:T\to\mathbb{G}_{m}$. This cocharacter is called a coroot. We denote the set of coroots by $\Phi^{\vee}$.

The datum $(\Phi, X^{*}(T), \Phi^{\vee}, X_{*}(T))$ is called the root datum associated to $G$. This root datum is actually independent of the chosen maximal torus, which follows from all maximal tori being contained in a unique conjugacy class in $G$.

There is also a concept of an “abstract” root datum, a priori having seemingly nothing to do with reductive groups, just some datum $(M, \Psi, M^{\vee}, \Psi^{\vee})$ where $M$ and $M^{\vee}$ are finitely generated abelian groups, $\Psi$ is a subset of $M\setminus \lbrace 0\rbrace$, and $\Psi^{\vee}$ is a subset of $M^{\vee}\setminus \lbrace 0\rbrace$, and they satisfy the following axioms:

• There is a perfect pairing $\langle,\rangle:M\times M^{\vee}\to\mathbb{Z}$.
• There is a bijection between $\Psi$ and $\Psi^{\vee}$.
• For any $\alpha\in \Psi$, and $\alpha^{\vee}$ its image in $\Psi$ under the aforementioned bijection, we have $\langle \alpha,\alpha^{\vee}\rangle=2$.
• For any $\alpha\in \Psi$, the automorphism of $M$ given by $x\mapsto \alpha-\langle x,\alpha^{\vee}\rangle\alpha$ preserves $\alpha$.
• The subgroup of $\mathrm{Aut}(M)$ generated by $x\mapsto \alpha-\langle x,\alpha^{\vee}\rangle\alpha$ is finite.

Again, a priori, such a datum of finitely generated abelian groups and their subsets, satisfying these axioms, seems to have nothing to do with reductive groups. However, we have the following amazing theorem:

Any abstract root datum is the root datum associated to some reductive group.

For reductive groups over an algebraically closed field, the root datum classifies reductive groups:

Two reductive groups over an algebraically closed field have the same root datum if and only if they are isomorphic.

Let us now discuss how root datum helps us understand the automorphisms of a reductive group. For this we need to expand the information contained in the root datum.

A root basis is a subset of the roots such that any root can be expressed as a unique linear combination of the roots, where the integer coefficients are either all positive or all negative. A based root datum is given by $(\Phi, X^{*}(T), S, \Phi^{\vee}, X_{*}(T), S^{\vee})$, i.e. the usual root datum together with the additional datum of a root basis $S$.

The root datum already determines the reductive group $G$. What does the additional data of a root basis mean? The root basis corresponds to a Borel subgroup of $G$ that contains our chosen maximal torus $T$. A Borel subgroup of $G$ is a maximal connected solvable Zariski closed algebraic subgroup of $G$.

A pinning is the datum $(G,T,B,\lbrace x_{\alpha}\rbrace_{\alpha\in S})$ where $T$ is a maximal torus, $B$ is a Borel subgroup containing $T$, and $\lbrace x_{\alpha}\rbrace_{\alpha\in S}$ is a basis element of $\mathfrak{g}_{\alpha}$. Given a pinning, a pinned automorphism of $G$ is an automorphism of $G$ that preserves the pinning.

An inner automorphism of a group $G$ is one that comes from conjugation by some element; in a way they are the automorphisms that are easier to understand. The inner automorphisms form a normal subgroup $\mathrm{Inn}(G)$ of the group of automorphisms $\mathrm{Aut}(G)$, and the quotient $\mathrm{Aut}(G)/\mathrm{Inn}(G)$ is called $\mathrm{Out}(G)$. We have similar notions for algebraic groups.

Now a pinned automorphism is an automorphism, therefore has a map to $\mathrm{Out}(G)$. A pinned automorphism also has a map to the automorphisms of the corresponding based root datum. Both of these maps are actually isomorphisms! Therefore we have a description of $\mathrm{Aut}(G)$ as follows:

The automorphisms of $G$ as an algebraic group are given by the semidirect product of the inner automorphisms and the automorphisms of the based root datum.

In this post we have only focused on the case of reductive groups over algebraically closed fields. Over more general fields the theory of reductive groups, for instance the classification, is more complicated. This will hopefully be tackled in future posts on this blog.

References:

Algebraic group on Wikipedia

Linear algebraic group on Wikipedia

Reductive group on Wikipedia

Root datum 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)

Lectures on the geometry and modular representation theory of algebraic groups by Geordie Williamson and Joshua Ciappara

Algebraic groups by J. S. Milne