A natural bijection between the orbit spaces of $p$-nilpotent matrices for varying $p$

Question

Let $k$ be an algebraically closed field of characteristic $p$, call a matrix $X\in\mathfrak{gl}_n(k)$ $p$-nilpotent if $X^p=0$, and let $\mathcal{N}_1=\mathcal{N}_1(\mathfrak{gl}_n(k))$ be the set of all $p$-nilpotent matrices, on which $\mathrm{GL}_n(k)$ acts by conjugation. Each $\mathrm{GL}_n(k)$-orbit of $\mathcal{N}_1$ is described by its Jordan form, which is a direct sum of blocks with eigenvalue zero, no block having size larger than $p$. In other words, the orbits correspond to partitions of $n$ with parts not greater than $p$.

Now, if we include the condition that $p\ge n$, we see that a description of the orbit space is independent of $p$, as all partitions of $n$ will have parts not greater than $p$. The point is that for $p\ge n$, nilpotent and $p$-nilpotent are equivalent conditions. It follows that if $k'$ is another algebraically closed field of characteristic $p'\ge n$, $p'\ne p$, there is a natural bijection between the orbits of $\mathcal{N}_1(\mathfrak{gl}_n(k))$ and those of $\mathcal{N}_1(\mathfrak{gl}_n(k'))$. We've arrived at this bijection by obtaining some combinatorial description of the orbit spaces, but I suspect there is something geometrical happening below the surface.

Without reference to Jordan forms or partitions, why do we have such a bijection of orbits for large characteristic?

One idea I've had is to consider $\mathrm{GL}_n$ as an affine group sceme, and $\mathcal{N}$ as an affine scheme, both defined over $\mathbb{Z}$. Here $\mathcal{N}$ is the full nullcone, consisting of all nilpotent matrices, whose $k$-points for $\mathrm{char}\;k\ge n$ are exactly $\mathcal{N}_1(k)$ as discussed above. Then, using the group scheme action $\mathrm{GL}_n\times\mathcal{N}\to\mathcal{N}$, we might be able to relate orbits of $k$-points with orbits of $k'$-points via the maps $\mathbb{Z}\to k$, $\mathbb{Z}\to k'$ and the given action on $\mathbb{Z}$-points. However, the relationship isn't very well-behaved. For example, consider the following non-conjugate, nilpotent matrices in $\mathrm{GL}_2(\mathbb{Z})$:

$$\begin{pmatrix}0&p\\0&0\end{pmatrix},\;\begin{pmatrix}0&p'\\0&0\end{pmatrix}$$

These matrices remain non-conjugate in characteristics $p$ and $p'$, but in all other characteristics they are conjugate. This behavior doesn't give me much hope for pursuing this idea further.

I want to generalize this bijection of orbits for large $p$ in the setting of $r$-tuples of pairwise-commuting nilpotent matrices. For $r>1$, I'm not aware of any combinatorial description of the orbits, so I'm looking elsewhere for a proof. Thanks in advance for any ideas concerning what phenomena are at work here.

Answer 1

This question is relatively old and not precisely enough formulated, so maybe it would help to clarify the questions actually being asked.

1) The header is misleading, since "$p$-nilpotent" isn't really at issue here. Instead, the prime $p$ is always taken to be large enough so that $p$-nilpotent is the same as nilpotent.

2) The initial question is about what happens to the variety of nilpotent elements (and its finitely many orbits) when the prime $p$ varies. Starting with $\mathrm{GL}_n$ (or equally well with $\mathrm{SL}_n$), it seems artificial to look for methods that ignore the usual role of partitions or Jordan decomposition in parametrizing the nilpotent orbits. On the other hand, for an arbitrary simple algebraic group $G$ over an algebraically closed field of good characteristic, the behavior of orbits is known to be uniform in all such characteristics. Some aspects of this have been studied only case-by-case, so it's always a challenge to improve on the existing methods. (For example, devising a uniform proof that the number of nilpotent orbits is finite in good characteristic is already difficult, and there seems to be no uniform proof yet for such a finiteness theorem in bad characteristic even though it turns out to be true in all cases.)

3) Apparently your main motivation comes from the study of commuting varieties or their generalizations. By definition, the commuting variety in $\mathfrak{g}$ consists of pairs of elements with zero commutator, which specializes in your situation to pairs of commuting nilpotent elements. Working with commuting $r$-tuples is then a natural generalization. To study such varieties, it would be best to consider all available techniques of algebraic geometry, combinatorics, and representation theory rather than to exclude any tool.

There is already a lot of literature on commuting varieties, first in characteristic 0 and more recently in prime characteristic. Samples include published papers by Premet here and by Nguyen here. As Nguyen's papers illustrate, interaction with some of the associated representation theory and cohomology of $G$ or $\mathfrak{g}$ may be suggestive.

4) Going back to your comparison of different (good) primes $p, p'$, probably the most natural approach (if it works) is to compare each of these with characteristic 0 rather than to try for any direct comparison. Just as with the older problems involving classification of orbits, it makes sense to refer back to classical results in characteristic 0 and confirm that these survive modulo $p$ or $p'$. How to do this effectively in terms of schemes isn't clear to me, when bad primes are possible too. In any case, all sorts of tools may turn out to be relevant, so I'd keep an open mind. The first step is of course to formulate more precisely what you expect to be true in terms of known results in characteristic 0.

While I can't answer your "bijection" questions directly, it does seem sensible to start with the existing literature.

Answer 2

Suppose $X$ is a variety over an algebraically closed field $k$ and $G$ is an algebraic group acting on $X$. Then there is a dichotomy:

1. There are finitely many orbits. In this case $X(k)/G(k)$ does not change when we change $k$ to a bigger algebraically closed field.

2. There are infinitely many orbits. In this $X(k)/G(k)$ does change when we change $k$ to a bigger algebraically closed field.

Maybe this is really what you wanted to say in your question, I'm not sure. To prove it look at a flattening stratification for the stabilizer group scheme over $X$ and use a theorem on existence of quotients. (Presumably there is an elementary proof as well.)

Now for the final question: when looking at conjugacy classes of $r$-tuples of commuting nilpotent elements we will get infinitely many conjugacy classes pretty soon (I think this happens already for r = 2 and dimension 2).