Steinberg's lecture at the 1966 ICM in Moscow here surveyed his work on regular elements of semisimple algebraic groups, while also formulating a number of then-open questions as "problems" (with positive solutions expected). For example, he wrote: "A more modest problem is as follows. (10) Problem. Prove that the number of unipotent classes of $G$ is finite." [This was later proved by Lusztig, using an ingenious argument based on Deligne-Lusztig character theory for Chevalley groups; the problem is not so "modest".] Another question (22) was later disproved. But by now most of the problems have well-documented outcomes. In one case I don't recall any relevant sources, which prompts my question:
Is Steinberg's Problem (12), described below, ever referred to in the subsequent literature?
To fill in essential background here (using Steinberg's notation), take $G$ to be a connected semisimple and simply connected algebraic group over an algebraically closed field $K$, with Lie algebra $L$. It's enough to assume $G$ is simple. [Here $L$ is in fact the Lie algebra over $K$ obtained by Chevalley's process from a Chevalley basis in a simple Lie algebra of the same type over $\mathbb{C}$.]
Now Steinberg considers the adjoint representation of $G$ on $L$, with $G_x$ denoting the centralizer of $x$ in $G$; its Lie algebra Lie($G_x$) is contained in the subalgebra $L_x$ of $L$ fixed pointwise by Ad $x$, as is the center of $L$. When $p$ is good (leaving aside type $A$), Steinberg shows in (11) that $L_x$ is just the Lie algebra of $G_x$. In (12) he asks for a proof that $L_x$ is the sum of Lie($G_x$) and the center of $L$ for all $p$ and all $G$, noting that the case of regular $x$ follows from his own study of regular elements. [Recall that $p$ is bad if $p=2$ for types other than $A$, or $p=3$ for exceptional types, or $p=5$ for type $E_8$.].
It turns out that (12) sometimes fails. One example, pointed out by Sasha Premet in a 2015 email, involves the unique group $G$ of type $E_8$ and the bad prime $p=5$. It's been known, since early work by Dieudonne and also by Steinberg, that $L$ here is simple for all $p$ (and thus is centerless). For $p=5$, the 1980 paper of Mizuno shows that the list of unipotent classes and their dimensions is the same as for p=0. For the class labelled $A_4+A_3$ by Bala-Carter, the dimension of $G_x$ is therefore 48. On the other hand, 1995 computations by Lawther in his Comm. Algebra paper show that in the smallest faithful representation of the group of type $E_8$ (the adjoint representation) one gets $\dim L_x=50$ (Table 9, p. 4148). This follows from Lawther's computation of Jordan blocks for a conjugate of $x$: the number of blocks is the number of independent fixed points of Ad $x$. Because $L$ is centerless, this disagrees with Steinberg's expectation in (12).
[EDIT] I'm still curious as to whether there is any follow-up to that problem in the literature.
