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Define $G(n, k)$ as a subgroup of $\rm{Aut}(\Bbb C[[x_1, \dots, x_n]]/\mathfrak m^{k+1})$ with identity linear part (so, group of $k$-jets of selfmaps of $\Bbb C^n$). I'm interested in the map from $G(2, k)$ to the set of its conjugacy classes. To be more precise, there are two subquestions.

  1. What is the space of conjugacy classes?

  2. Is there some reasonable combinatorial system describing fibers of that map?

I'd be happy to know anything about it in case of $\Bbb C$ being replaced by finite field as well, not sure if it's harder or easier. There are some results on enumerating conjugacy classes in triangular groups, for example, there's only finite number of centralizers orbits for triangular matrices of size $<6$, and Kolchin thm may be helpful a bit.

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    $\begingroup$ I guess that $\mathfrak{m}$ is the maximal ideal at $0$ (which would be more obvious using $\mathbf{C}[[x_1,\dots,x_n]]$). I also guess that $\mathrm{Con}(G)$ means "set of conjugacy classes of the group $G$", and that "a map" should be "the map". Anyway you're asking about conjugacy classes in a certain familiy of explicit unipotent $\mathbf{C}$-groups. $\endgroup$
    – YCor
    Commented Oct 16, 2020 at 18:01
  • $\begingroup$ Yes, groups are very explicit, but somewhat complicated. Sorry, I have some trouble writing things clearly and exactly as I wanted. $\endgroup$
    – Denis T
    Commented Oct 16, 2020 at 18:13

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