Reference Request: Derived group of $\mathscr R_u(B)$

Question

Let $G$ be a connected, reductive group over an algebraically closed field $k$. Let $B$ be a Borel subgroup with maximal torus $T$ and unipotent radical $U$. Let $\Phi^+ = \Phi(B,T)$ and $\Delta$ the base of $\Phi = \Phi(G,T)$ corresponding to $\Phi^+$.

A subset $\Psi$ of $\Phi^+$ is called closed if whenever $\alpha, \beta \in \Psi$, and $\alpha + \beta$ is a root, we have $\alpha + \beta \in \Psi$. If $\Psi$ is closed, then the root subgroups $U_{\alpha} : \alpha \in \Psi$ directly span a closed, connected subgroup $U_{\Psi}$ of $U$ which is normalized by $T$ and whose Lie algebra is $\bigoplus\limits_{\alpha \in \Psi} \mathfrak g_{\alpha}$.

Furthermore, if $\alpha \in \Phi^+$, and $\Psi \subseteq \Phi^+$ is closed, and all roots of the form $a \alpha + b \Psi$ for $a, b \in \mathbb{Z}^+$ lie in $\Psi$, then the root subgroup $U_{\alpha}$ normalizes $U_{\Psi}$. These facts are proved in Chapter 14 of Borel, Linear Algebraic Groups.

In particular, let $\Psi = \Phi^+ - \Delta$. It is easy to see that $\Psi$ is closed and normalized by every root subgroup $U_{\alpha} : \alpha \in \Phi^+$, and in particular, $U_{\Phi^+ - \Delta}$ is a normal subgroup of $U$. Moreover, it is a consequence of 8.32 in Springer's Linear Algebraic Groups that for any $x \in U_{\alpha}$ and $y \in U_{\beta}$, the commmutator $xyx^{-1}y^{-1}$ lies in $U_{\Phi^+ - \Delta}$. From here one can argue that $U_{\Phi^+ - \Delta}$ contains the derived group of $U$ by producing a homomorphism $U \rightarrow \prod\limits_{\alpha \in \Delta} \mathbf G_a$ with kernel $U_{\Phi^+ - \Delta}$.

My question is, is $U_{\Phi^+ - \Delta}$ exactly the derived group of $U$? For $G = \textrm{GL}_n$, this does seem to be the case.

Answer 1

I think the answer is yes.

We can assume that $G$ is semisimple. As you have seen already, and which is is clear from the commutation relations, we have $[U, U] \leq \prod_{\alpha \in \Phi^+ - \Delta} U_\alpha$. I guess you could show equality by using the Chevalley commutator relations, although you need to be careful when $p$ is small (e.g. $G$ of type $B_n$ and $p = 2$).

Here is an idea for a different way to see this. The containment $[U, U] \leq \prod_{\alpha \in \Phi^+ - \Delta} U_\alpha$ shows that $$\dim U / [U, U] \geq \dim U / \prod_{\alpha \in \Phi^+ - \Delta} U_\alpha = \operatorname{rank} G$$

So it would be enough to show that $\dim U/[U,U] \leq \operatorname{rank} G$.

For this, a result of Steinberg states that there exists a regular unipotent element $u \in G$, which has the property that $\dim C_G(u) = \operatorname{rank} G$. One can show that $u$ is contained in a unique Borel subgroup $B$ of $G$, and that $C_G(u) = C_B(u) = C_U(u)Z(G)$, where $U$ is the unipotent radical of $B$. Proofs can be found in reference (*) below.

In any case, we have a unipotent element $u \in U$ such that $\dim C_U(u) = \operatorname{rank} G$. The conjugacy class $\operatorname{Cl}_U(u)$ of $u$ in $U$ is contained in the coset $u[U,U]$ (proof: $g^{-1}ug = u[u,g]$), so $$\dim \operatorname{Cl}_U(u) = \dim U - \dim C_U(u) \leq \dim [U,U]$$

giving $\dim U/[U,U] \leq \operatorname{rank} G$.

(*) See Chapter 4, sections 4.1 - 4.5 in

Humphreys, James E.: Conjugacy classes in semisimple algebraic groups. Mathematical Surveys and Monographs, 43. American Mathematical Society, Providence, RI, 1995.

or:

Theorem 3.1 in:

Steinberg, Robert: Regular elements of semisimple algebraic groups. Inst. Hautes Études Sci. Publ. Math. No. 25 1965 49–80.

and Lemma 4.3 in:

Springer, T. A.: Some arithmetical results on semi-simple Lie algebras. Inst. Hautes Études Sci. Publ. Math. No. 30 1966 115–141.

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PS. For further terms of the derived series and the lower central series, you can take a look at the following paper, where they are described when $p$ is not too small for $G$.

Azad, H.; Barry, M.; Seitz, G.: On the structure of parabolic subgroups. Comm. Algebra 18 (1990), no. 2, 551–562.

Answer 2

Let me add a few comments in community-wiki format. There doesn't seem to be a convenient reference, apart from the one in Digne-Michel which Jay Taylor cites. But even here, the authors don't give a full proof of the existence of regular unipotent elements. Such regular elements and their properties are essential to the approach Mikko gives, though it's possible that there is a more elementary method yet to be found. What's clear is that a considerable amount of structure theory for semisimple groups is involved. (Of course, in characteristic 0 one can instead work more straightforwardly in the Lie algebra.)

Steinberg's treatment of regular elements (IHES, 1965) is available online through numdam.org, as is Springer's article (IHES, 1966). But note that Springer didn't succeed for bad primes in arriving at a proof of existence for regular unipotents. What he did was more direct than Steinberg's method, relying mainly on Chevalley's basis and commutation formula. Later on, students of Steinberg pushed this technique further for bad primes, but it's unclear how to extract a uniform theoretical approach. (Recently I wrote up some notes attempting to sort out the arguments used for both regular unipotents and regular nilpotents, posted here.)

As nsfc23 comments, it's tricky to work directly with Chevalley's commutator formula in some small characteristic cases. On the other hand, Steinberg's approach to regular unipotents requires fairly heavy machinery. As to finiteness of the number of unipotent clases (still conjectural in the mid-1960s), it remains an open problem to use modular representation theory of $G$ or its Lie algebra to get a more self-contained proof.