Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$-orbits in ${\mathfrak g}$ and that the classification of these orbits is the same as over the complex numbers, as long as the characteristic of $k$ is *good*, i.e. odd if $G$ is not of type $A$, greater than $3$ if $G$ is of exceptional type, and greater than $5$ if $G$ is of type $E_8$. There is quite a lot of history to the subject, but a <a href="http://core.kmi.open.ac.uk/download/pdf/286713.pdf">uniform proof</a> of the classification was given relatively recently by Premet.

My question concerns the closure ordering on nilpotent classes: where (if anywhere) is it established that the closure ordering is also (in good characteristic) the same as over the complex numbers? This fact seems to be used in *Varieties of nilpotent elements for simple Lie algebras I: good primes* by the VIGRE group, but the reference is to Carter's book, and I can't find anywhere in the VIGRE paper where they mention a justification for why the order should be the same. (It may be in there, but I haven't found it.)

For classical Lie algebras outside characteristic 2, I think it is reasonably straightforward to use the partition type classification to show that the closure order is also independent of the characteristic. So this is really a question about exceptional Lie algebras.