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 $m<n$).

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?

  • $\begingroup$ 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? $\endgroup$ – AHusain Jul 22 '19 at 7:12
  • $\begingroup$ 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 $G_a$. $\endgroup$ – Victor Protsak Jul 22 '19 at 8:24
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    $\begingroup$ 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). $\endgroup$ – Victor Protsak Jul 22 '19 at 8:35

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