Closure order on nilpotent orbits in exceptional Lie algebras

Question

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 uniform proof 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.

Answer 1

I'm assuming your question involves just good prime characteristic $p$. Much of the literature focuses on unipotent classes, but Springer's $G$-equivariant isomorphism (for good $p$) between the unipotent variety in $G$ and the nilpotent variety in $\mathfrak{g}$ shows that the classes and orbits are in bijection and also allows one to transfer the closure relationships. Thus the closure ordering graphs are the same for the group and the Lie algebra. (Here one has to be a bit careful about the isogeny type of $G$ in type $A_n$, however.)

It turns out after some work that the closure orderings of unipotent classes are the same as those found much earlier by Gerstenhaber and others in characteristic 0. Much of this work was done by Spaltenstein (and those he cites including Mizuno, Shoji for exceptional types): see his Classes unipotentes et sous-groupes de Borel, Lect. Notes in Math. 946 (Springer, 1982), especially II.8 and the graphs for exceptional types in IV.2. [Carter's 1985 book follows this development, though for types $E_7, E_8$ on pages 442 and 444 the graphs lack several edges; this was probably a technical error made during the production of the book.] Though I've never checked all the details carefully, I've been assured that experts have done so and find Spaltenstein's results convincing.

ADDED: Maybe it's useful to expand this answer and also respond to Ma's question. I should emphasize that the classification of nilpotent orbits or unipotent classes requires a lot of case-by-case work for each simple, simply connected algebraic group and its Lie algebra (obtained by Chevalley's process from a $\mathbb{Z}$-form of the corresponding simple Lie algebra over $\mathbb{C}$). Much of the literature here follows Chevalley's viewpoint, in which the unipotent classes are found to have representatives specified by certain parameters independent of good characteristic. (The bad primes, possibly $2,3,5$, may divide some of Chevalley's structure constants.) In particular, it's enough to work over finite fields of good characteristic, or over $\mathbb{Q}$.

For exceptional Lie types, only $G_2$ can be done in a fairly direct way. For $F_4$ the unipotent classes in finite Chevalley groups were studied by Shoji. Then Mizuno made very detailed computations for types $E_6, E_7, E_8$. Spaltenstein drew the literature together in his work on the Springer correspondence, providing many refinements and a few corrections as well as an emphasis on the hidden duality of special classes which leads to a symmetry for these classes in the closure order diagram (generalizing the classical partial ordering of partitions in type $A_\ell$).

The closure ordering itself is a relatively elementary byproduct of the classification when done in this spirit: roughly speaking, each parameter involved in a particular class is specialized to 0 and leads to other classes in the closure. But the classification itself is quite arduous to work out in detail as Shoji and Mizuno did. (It is probably best organized theoretically in the algebraic group context by the Bala-Carter method.)