Suzuki and Ree groups, from the algebraic group standpoint

Question

The Suzuki and Ree groups are usually treated at the level of points. For example, if $F$ is a perfect field of characteristic $3$, then the Chevalley group $G_2(F)$ has an unusual automorphism of order $2$, which switches long root subgroups with short root subgroups. The fixed points of this automorphism, form a subgroup of $G_2(F)$, which I think is called a Ree group.

A similar construction is possible, when $F$ is a perfect field of characteristic $2$, using Chevalley groups of type $B$, $C$, and $F$, leading to Suzuki groups. I apologize if my naming is not quite on-target. I'm not sure which groups are attributable to Suzuki, to Ree, to Tits, etc..

Unfortunately (for me), most treatments of these Suzuki-Ree groups use abstract group theory (generators and relations). Is there a treatment of these groups, as algebraic groups over a base field? Or am I being dumb and these are not obtainable as $F$-points of algebraic groups.

I'm trying to wrap my head around the following two ideas: first, that there might be algebraic groups obtained as fixed points of an algebraic automorphism that swaps long and short root spaces. Second, that the outer automorphism group of a simple simply-connected split group like $G_2$ is trivial (automorphisms of Dynkin diagrams mean automorphisms that preserve root lengths).

So I guess that these Suzuki-Ree groups are inner forms... so there must be some unusual Cayley algebra popping up in characteristic 3 to explain an unusual form of $G_2$. Or maybe these groups don't arise from algebraic groups at all.

Can someone identify and settle my confusion?

Lastly, can someone identify precisely which fields of characteristic $3$ or $2$ are required for the constructions of Suzuki-Ree groups to work?

Answer 1

It is not really a question of inner forms. What happens is that the algebraic group $G_2$ has an extra endomorphism $\varphi$ whose square is the Frobenius map (over the appropriate finite field). Just as for any algebraic group over a finite field $F$ its rational points over $F$ are the fixed points of the Frobenius endomorphism the Suziki groups are, by definition, the fixed points of $\varphi$. Again, just as the Frobenius, on points over the algebraic closure of $F$ $\varphi$ is an automorphism of the abstract group. However, that is misleading, the essential points is that it is an endomorphism (which definitely is not an automorphism) of the algebraic group. Most of the properties of points over $F$ of a semi-simple algebraic group $G$ defined over $F$ follows from the algebro-geometric theory of $G$ and the properties of the Frobenius endomorphism. Similarly, most of the properties of Suziki groups follows from the algebro-geometric theory of $G_2$ together with the properties of $\varphi$. As $\varphi$ is very similar to the Frobenius endomorphism this works almost the same way as if $\varphi$ were indeed a Frobenius endomorphism.

Addendum: As one simple example of the similarity of $\varphi$ to a Frobenius consider the problem of computing the order of the Suzuki groups. As the square of $\varphi$ is the Frobenius, the action of it on the tangent space at any fixed point is nilpotent. This implies that such a fixed point appear with multiplicity one in the Lefschetz fixed point formula and the order of its group of fixed points is thus equal to the Lefschetz trace on (étale) cohomology of the algebraic group $G_2$. That cohomology can be canonically expressed in terms the action of the Weyl group on the character group of the maximal torus (see for instance example in SGA 4 1/2) and how $\varphi$ acts on that character group is essentially part of the definition of $\varphi$.

Answer 2

To supplement Torsten's account, the original Suzuki groups of type $C_2$ in characteristic 2 resulted from a purely group-theoretic investigation but were then recovered in the algebraic group setting. The Ree groups of types $F_4, G_2$ in respective characteristics 2, 3 were constructed inside the Chevalley groups of these types but also recovered in a uniform way by Steinberg in Endomorphisms of algebraic groups (AMS Memoir). There is also a full account in my recent LMS Lecture Note volume Modular Representations of Finite Groups of Lie Type (Cambridge, 2006). The algebraic group viewpoint is outlined by Torsten. The Suzuki and Ree groups don't arise from the split vs. quasisplit classification over finite fields, but rather involve Chevalley's special isogenies which interchange root lengths while using a finite field automorphism. The orders of the finite fields one starts with are the odd powers of 2, 3 respectively. But notation is tricky, since some people like to express things in terms of square roots to make the finite group orders resemble those of the corresponding split groups.

Since the Suzuki and Ree groups have BN-pairs, it is popular with finite group theorists to use this viewpoint in studying them (simplicity, etc.).

Answer 3

The second part of Karsten Naert's thesis constructs Suzuki—Ree groups as groups of points of "algebraic groups over $\mathbb{F}_{\!\sqrt{p}}$". Such fields being nonexistent, these "algebraic groups" are understood not as schemes, but as the so-called "twisted schemes". They live in the category of pairs $(X, \Phi_X)$, where $X$ is a scheme and $\Phi_X\in \operatorname{End}(X)$ satisfies $\Phi_X\circ\Phi_X=F_X$, the Frobenius morphism.

Another way to describe Suzuki—Ree groups as something "defined by equations" (not algebraic, but difference-algebraic) is to compare two fundamental representations of the ambient Chevalley group in characteristic 2 or 3, whose highest weights $\lambda$ and $\mu$ are interchanged by the exceptional symmetry. These representations happen to have the same dimension, so one considers the group of all $g$ such that $\tau g_ \lambda=g_\mu\tau$, where $\tau^2$ is the Frobenius morphism. This produces the Suzuki—Ree groups, requires no generators and relations and works over any ring of characteristic $p$ admitting such $\tau$ (in particular, the field of definition need not be perfect).