# cohomology of nilpotent matrices of fixed $m$-th power

Let $$k$$ be an algebraically closed field, $$\mathcal{N}$$ is the variety of $$n \times n$$ nilpotent matrices over $$k$$, and consider the natural $$m$$-power map $$\mathcal{N} \rightarrow \mathcal{N}$$ given by sending $$A$$ to $$A^m$$ (we assume $$m).

I am intersted in the fibers of this map, the preimage of zero is well-studied, how about other fibers (dimension, irreducible components, singularity)? What are their compactly supported cohomology groups?

• For some of the numerical invariants, have you calculated them as functions of partitions and then enter them at findstat.org to see if it matches with something combinatorial? – AHusain Jul 22 '19 at 7:12
• Is this really the question you want to study? There is a crucial difference between the preimage of zero and other fibers. Namely, this map $F$ is equivariant under the conjugation by $G={\rm GL}_n$ and $0$ is the unique fixed point. So unlike $F^{-1}(0)$, which is invariant under a reductive group $G$ (in fact, it is the closure of a single orbit of $G$), the other fibers $F^{-1}(a)$ are invariant only under a smaller non-reductive group $G_a=C_G (a)$. While $G$-action on $\mathcal N$ has a dense open orbit, and in fact only finitely many orbits, this is not the case for $G_a$. – Victor Protsak Jul 22 '19 at 8:24
• A natural and more manageable question is to describe the preimage $F^{-1}(X)$, where $X$ is an irreducible $G$-invariant subvariety of $\mathcal N$, for example a nilpotent orbit or its closure. This preimage is $G$-invariant. One can then try to extract the information about $F^{-1}(a)$ for various $a\in X$ (if $X$ is an orbit, then these varieties are isomorphic by the $G$-action). – Victor Protsak Jul 22 '19 at 8:35