reductive groups

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

Theories and Theorems

Math and Physics for Everyone

Reductive Groups Part I: Over Algebraically Closed Fields