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This question needs some background:

(1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be replaced by any algebraically closed field of characteristic 0).

In particular, he looked at a $3$-dimensional simple subalgebra with basis $(e,f,h)$ for which $e$ is a principal (= regular) nilpotent element. Such elements are characterized by having centralizer in $\mathfrak{g}$ of smallest dimension, equal to the rank $\ell$. In turn these nilpotents form a dense open orbit in the nullcone $\mathcal{N}$ of $\mathfrak{g}$ under the action of the associated adjoint group $G$ having Lie algebra $\mathfrak{g}$. For example, the sum of root vectors for a fixed set of simple roots is regular nilpotent.

If $q$ denotes the height of the highest positive root (the sum of coefficients of this root when written as a $\mathbb{Z}^+$-linear combination of simple roots), then $q+1$ was seen to be the Coxeter number $h$ of the Weyl group. Kostant proved that $e$ is regular iff $(\mathrm{ad} \,\; e)^{2q} \neq 0$. On the other hand, $(\mathrm{ad} \, e)^{2q+1} = (\mathrm{ad} \, e)^{2h-1} = 0$, which by density carries over to all nilpotents $e$. [Note the obvious misprint in Kostant's Cor. 5.4.)

The classification gives respective values $h = \ell+1,\, 2 \ell,\, 2\ell, \,2\ell -2,\, 12, 18, 30, 12, 6$ in types $A_\ell, \, B_\ell, \, C_\ell, \, D_\ell, \, E_6, \, E_7, \, E_8, \, F_4, \, G_2$.

(2) Much of this carries over to an adjoint group $G$ and its Lie algebra $\mathfrak{g}$ over an algebraically closed field of characteristic $p>0$, which is restricted relative to a $[p]$-operation. We assume $p$ is good (doesn't divide any coefficient of the highest root), which is always true in type $A_\ell$ but requires avoidance of primes $2,3,5$ in some other types. Then $\mathfrak{g}$ is simple or close to simple, and its nullcone $\mathcal{N}$ is a union of finitely many $G$-orbits just as in characteristic 0. In particular, there is still a dense orbit of regular nilpotents, and the sum of root vectors for simple roots is again regular: see $\S5$ in Springer's 1966 paper here:.

(3) In the mid-1980s, Janzen and Friedlander-Parshall independently developed properties of cohomological support varieties for (restricted) $\mathfrak{g}$-modules, parallel to the theory for finite groups developed by Carlson and others. Here support varieties are closed conical subvarieties of $\mathcal{N}$ lying in $\mathcal{N}_1:= \{e \in \mathcal{N}\: | \: e^{[p]}=0\}$.

In general, each $e \in \mathcal{N}$ satisfies $e^{[p^r]}=0$ for some $r>0$. Jantzen observed in 4.1 here that when $p \geq h$, one always has $\mathcal{N}_1 = \mathcal{N}$. In turn, $(\mathrm{ad} \, e)^p=0$ for all $e \in \mathcal{N}$. Though his brief remarks don't give a complete proof, the idea is to prove this for regular nilpotents $e$ and then invoke density: Use a 1-dimensional torus in $G$ (via the sum of positive coroots) whose nonzero eignevalues on $\mathfrak{g}$ are twice the heights of roots. Now $e$ has eigenvalue 2 and $e^{[p]}$ has eigenvalue $2p$; so Kostant's characterization of $h$ comes into play.

(4) In one of its computation-oriented projects, the U. Georgia VIGRE Algebra Group here computed for all good $p$ which powers of nilpotents are 0. In partcular, their tables confirm Jantzen's observation for $p \geq h$ while also showing some divergence in characteristic 0 between his exponent and Kostant's result.

Example: In type $E_8$, the Lie algebra $\mathfrak{g}$ is simple for all $p$ and its adjoint representation is also its smallest faithful representation (of dimension $248$). Here Kostant only gets $(\mathrm{ad} \, e)^{59}=0$ for all nilpotents $e$, whereas in prime characteristics $p=31, 37, \dots, 53$ one already gets vanishing of $p$th powers.

Probably it's possible in case-by-case computation, using a Chevalley basis of $\mathfrak{g}$, to see how the integral coefficients accumulate for powers of $\mathrm{ad} \, e$ when $e$ is as above a special type of regular nilpotent and the operator is applied first to a lowest root vector: then $p$ should divide all coefficients after applying it $p$ times. But I wonder if there is a classification-free proof:

Is there a uniform explanation for this difference in the minimal vanishing power of $\mathrm{ad} \, e$, in characteristic 0 and in good prime characteristic?

[Minor edit to correct a couple of my own misprints.]

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Let $T$ be a maximal torus of $G$ and $\Delta$ a basis of simple roots relative to $T$. As you point out, $e=\sum_{\alpha\in\Delta}e_\alpha$ is a regular nilpotent element. For any root $\beta=\sum_{\alpha\in\Delta}n_\alpha \alpha$ the height of $\beta$ is $\sum_{\alpha\in\Delta} n_\alpha$. Then we have a grading of ${\mathfrak g}$ by root-heights: let ${\mathfrak g}(0)={\rm Lie}(T)$ and for $i\neq 0$ let ${\mathfrak g}(i)$ be the span of all root subspaces ${\mathfrak g}_\alpha$ where $\alpha$ has height $i$. In particular $e\in{\mathfrak g}(1)$ and the highest non-trivial part of the grading is (in your notation) ${\mathfrak g}(q)={\mathfrak g}_{\hat\alpha}$, where $\hat\alpha$ denotes the highest root.

Now let $x\mapsto x^{[p]}$ denote the $p$-operation on ${\mathfrak g}$. Then $e^{[p]}\in{\mathfrak g}(p)$. In particular, if $p\geq h$ then $e^{[p]}=0$. By very standard properties of the $p$-operation, this means that $({\rm ad}\, e)^p = {\rm ad}\, e^{[p]}=0$.

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  • $\begingroup$ Yes, this is a more standard version of the argument I sketched; but I tend in char p to focus on gradings arising from 1-dimensional tori, which people including McNinch and Testerman have used in more sophisticated ways to bypass char 0 methods such as Jacobson/Morozov embeddings. But I'm still puzzled by the highlighted question about the contrast in vanishing results as the characteristic changes. The usual reduction mod $p$ of a Chevalley basis connects these settings and gives the Lie algebra of a simply connected $G$. $\endgroup$ – Jim Humphreys May 6 '16 at 14:21
  • $\begingroup$ P.S. Though it's not directly related to the good prime framework here, a student of Steinberg named Sharad Keny did check case-by-case for bad $p$ in her 1978 thesis that the recipe here still gives a regular nilpotent element (the existence of such elements being then uncertain). $\endgroup$ – Jim Humphreys May 6 '16 at 14:28
  • $\begingroup$ Is there any relevant literature on the interaction of the $p$-map with the root lattice grading of $\mathfrak{g}$, for example stating explicitly that $e^{[p]} \in \mathfrak{g}(p)$? I mentioned the somewhat more elaborate framework used by McNinch-Testerman partly because it is well documented in their work. This is a side issue but helps to establish a standard conceptual argument for the equality $\mathcal{N} = \mathcal{N}_1$ when $p \geq h$. $\endgroup$ – Jim Humphreys Jul 4 '16 at 14:16

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