# Twisted root subgroups in twisted Chevalley groups (reference request)

I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups.

Let me first recall the classical set-up. According to Steinberg's lecture notes, every simple untwisted Chevalley group $$G$$ can be obtained as follows. Consdider a finite-dimensional, complex simple Lie algebra $$L$$ together with a Chevalley basis and a finite field $$\mathbb{F}$$. Associate to each root $$\alpha$$ a one-parameter subgroup $$x_\alpha : \mathbb{F} \longrightarrow \operatorname{GL}(L) : t \mapsto x_\alpha(t)$$, called the root subgroup of $$\alpha$$, in the usual way (using the exponential of the ad-nilpotent matrices, interpreted over the field $$\mathbb{F}$$). Then each such $$x_\alpha$$ is an injective homomorphism from the additive group $$(\mathbb{F},+)$$ of the field $$\mathbb{F}$$ to the group $$\operatorname{GL}(L)$$ and $$G$$ is (isomorphic to) the group that is generated by the elements $$x_\alpha(t)$$ (where $$\alpha$$ ranges over the roots and where $$t$$ ranges over the elements of the field). These root subgroups are moreover compatible with field automorphisms in the obvious way.

The case of twisted Chevalley groups is more complicated to describe. We assume that the Dynkin diagram and the field allow us to define a distinguished automorphism $$\sigma = \gamma \cdot \phi$$ of the group $$G$$ in the usual way (with $$\gamma$$ a symmetry of the Dynkin diagram and with $$\phi$$ a suitable field automorphism), so that we may define the twisted Chevalley group associated to $$G$$ and $$\sigma$$ as $$G^\sigma$$, the subgroup of $$G$$ that is fixed elementwise by $$\sigma$$.

I would now like to have a collection of injective homomorphisms $$y_B : (\mathbb{F},+) \longrightarrow G^\sigma : t \mapsto y_B(t)$$ that are compatible with field automorphisms and that generate $$G^\sigma$$, as in the classical case. Here, the field $$\mathbb{F}$$ is the defining field, (or possibly the subfield $$\mathbb{F}_0$$ of $$\mathbb{F}$$ that is fixed by the distinguished field automorphism $$\phi$$).

The obvious choice of $$x_\alpha$$ does not work, since the $$x_\alpha$$ need not be contained in $$G^\sigma$$. A natural, but slightly naive, approach would be to consider a $$\langle \gamma \rangle$$-orbit B and to then define

$$y_B : \mathbb{F}_0 \longrightarrow G^\sigma : t \mapsto x_\alpha(t)$$ (for an orbit $$B = \{ \alpha \}$$ of length $$1$$),

$$y_B : \mathbb{F} \longrightarrow G^\sigma : t \mapsto x_{\alpha}(t) \cdot x_{\gamma^1(\alpha)}(\phi^1(t))$$ (for an orbit $$B = \{ \alpha , \gamma(\alpha) \}$$ of length $$2$$),

$$y_B : \mathbb{F} \longrightarrow G^\sigma : t \mapsto x_{\alpha}(t) \cdot x_{\gamma^1(\alpha)}(\phi^1(t)) \cdot x_{\gamma^2(\alpha)}(\phi^2(t))$$ (for an orbit $$B = \{ \alpha , \gamma(\alpha) , \gamma^2(\alpha) \}$$ of length 3).

Question: Is there a standard text that constructs a collection of one-parameter subgroups $$t \mapsto y_B(t)$$ of $$G^\sigma$$ satisfying the following properties?

• Property 1: Each $$y_B$$ is an injective homomorphism.

• Property 2: The group $$G^\sigma$$ is generated by the elements $$y_B(t)$$.

• Property 3: The one-parameter groups $$y_B$$ are compatible with field automorphisms, in the sense that if the field automorphism $$A$$ is induced by the automorphism $$a$$ of the field $$\mathbb{F}$$, then $$A(y_B(t)) = y_B(a(t))$$ for all $$t \in \mathbb{F}$$.

Assuming that the $$y_B$$ are indeed defined using the above $$\langle \sigma \rangle$$-orbits, Property 1 would seem to follow from the "uniqueness of expression" in the product formulae for the classical case, while Property 3 would follow immediately.

Gorenstein and Lyons make a vague reference to such a construction in [The local structure of finite groups of characteristic 2 type: Part I, Chapter I, Paragraph 4]. It also looks like E. Abe has indeed tried "something along those lines" in [Coverings of twisted Chevalley groups over commutative rings. Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1977), no. 366-382, 194–218.] But the text is difficult to understand and I am still unsure about the conventions / assumptions / generality of it.

Any help would be appreciated.

• I don't know if this helps, but root subgroups of twisted Chevalley groups are discussed in the book 'Simple groups of Lie Type' by Roger. W. Carter. Commented Jan 28, 2022 at 10:24
• The book was indeed helpful. Thank you for the recommendation. Commented Feb 2, 2022 at 12:48

Tom's answer is the complete story with two kinds of exceptions: (a) $$G^\sigma$$ is a Suzuki-Ree-type group -- $${^2}B_2(2^a)$$, $${^2}F_4(2^a)$$, or $${^2}G_2(3^a)$$; or (b) there exists a root $$a_0$$ such that $$a_0+\gamma(a_0)$$ is again a root.

In case (a), the symmetry of order $$2$$ of the Dynkin diagram leads to an automorphism of $$G$$ which is not in general of order $$2$$, but can be composed with a suitable field automorphism to give the desired automorphism $$\sigma$$ of order $$2$$. This process is described in Steinberg's lecture notes.

In case (b), $$\sigma$$ arises as you describe, but Property 1 fails: $$\gamma_B$$ is not a homomorphism for the orbit $$B$$ containing $$a_0$$. This is an issue for $${^2}A_{2n}(q)$$.

Actually, case (a) is a subcase of case (b), but I've separated out the Suzuki-Ree-type groups since their construction has extra complications.

As a result there are actually 7 types of "twisted root groups," of which you have described three. Three of the other four are not abelian. They are all $$p$$-groups, where $$p$$ is the characteristic of the underlying fields.

A fourth type occurs in the odd-dimensional unitary groups, when $$a_0$$ and $$\gamma(a_0)$$ are fundamental roots in an $$A_2$$-subsystem of the root system of $$G$$. (You can see from the Dynkin diagram that there is at least one such $$a_0$$.) This type has nilpotence class $$2$$. A fifth type gives the Sylow $$2$$-subgroups of the Suzuki groups $${^2}B_2(q)$$, and also appears in the groups $${^2}F_4(q)$$. It too has class $$2$$, with one small exception. A sixth type is abelian and also occurs in $${^2}F_4(q)$$, and the seventh type, of class $$3$$ with one small exception, gives the Sylow $$3$$-subgroups of the Ree groups $${^2}G_2(q)$$.

This is discussed in some detail in Chapter 2, particularly in Section 2.4, of: D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups, Number 3. Part I, Chapter A: Almost Simple $$K$$-Groups. Mathematical Surveys and Monographs vol. 40, American Mathematical Society, 1998.

And Properties 2 and 3 remain true in all cases.

• I understand that my intuition was a little off here. Thank you for explaining the whole situation in such detail. The book was also more than helpful. Commented Feb 2, 2022 at 12:42

As indicated in Martin Seysen's comment, this construction can be found in Carter's book "Simple Groups of Lie type".

More precisely, this is Proposition 13.6.3, and your "naive approach" is indeed exactly what you need to do. Your "Property 2" is Proposition 13.6.5 in Carter's book. Property 1 is indeed obvious from the classical case —I'm not sure why you are hesitant in your formulation— and Property 3 is indeed immediate as well.