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 is at work here.

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.