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Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by conjugation.

  1. One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan blocks for $\operatorname{GL}_n$),
  2. There is an especially interesting class, of nilpotent orbits. There are finitely many, labelled by combinatorial data attached to $G$ (e.g. partitions of $n$), and are related to a lot of interesting maths.

My question is: what when you replace $\mathfrak{g}^*$ by a general finite dimensional representation $V$ (with some conditions, if you like)? What can be said about their orbits: are they interesting, and can they be classified?

Possibly the answer might just be no, and that $\mathfrak{g}^*$ is special because it's Poisson/has other special properties, but hopefully at least some other $V$'s are interesting.

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  • $\begingroup$ In terms of what data would you expect a classification to be described? Maybe something like a highest weight for $V$, if it is irreducible? (Also, in 'general', meaning outside characteristic $0$, it is the adjoint setting where one expects a nice characterisation of orbits; coadjoint orbits can be subtler in characteristic $p$. Of course, over $\mathbb C$, it doesn't matter!) $\endgroup$
    – LSpice
    Commented Mar 23, 2022 at 22:45
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    $\begingroup$ Also, the analogue of nilpotency in general representations is "($0$-)instability"; see Kempf - Instability in invariant theory. In turn, the analogue of semisimplicity is stability. I do not know of any results on a general "Jordan decomposition"; Kac and Weisfeiler (in their Indag. Math. article "Coadjoint orbits …", to which I cannot find a link right now) discuss it in the (positive-characteristic) coadjoint case. $\endgroup$
    – LSpice
    Commented Mar 23, 2022 at 22:50
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    $\begingroup$ Thank you for this. Yes, I would be happy to see a classification when $V$ is irreducible, in terms of its highest weight, and would hope that those analogues of nilpotent orbits you mentioned also have a classification of a similar flavour to the classification of nilpotent orbits by partitions. $\endgroup$
    – Pulcinella
    Commented Mar 24, 2022 at 14:07
  • $\begingroup$ I think hoping for a classification by partition-type objects might be too much; this is (as far as I know) not available even in the exceptional coadjoint case. Maybe, if one restricts to $G$ classical, then there is still hope? $\endgroup$
    – LSpice
    Commented Mar 24, 2022 at 14:14
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    $\begingroup$ I think even in the case of classical G and irreducible V there is no hope in general. For example, the action of $GL(U) \times GL(V) \times GL(W)$ on the irrep $U \otimes V \otimes W$ is the tensor isomorphism problem. It is "tensor wild" (harder than just representation-wild), TI-complete, harder than Graph Isomorphism. Classifying orbits there over F_p is equivalent to classifying class 2 p-groups of exponent p, a notoriously hard and thought-to-be-impossible problem. $\endgroup$
    – Joshua Grochow
    Commented Mar 24, 2022 at 14:34

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Generally they can be classified when the action of $G$ on $V$ is visible, which by definition means there are finitely many orbits in the nullcone. Irreducible visible pairs $(G, V)$ were classified in Kac - Some remarks on nilpotent orbits (with some corrections in Dadok–Kac - Polar representations). Most of them come from Vinberg's theta-groups, which are graded Lie algebras/groups (graded by the integers or integers mod $n$). The visible action in those cases is then to consider the adjoint action of the Lie group on its Lie algebra, restricted to the zero-graded piece $G_0$ of the group acting on the 1-graded piece $\mathfrak{g}_1$ of the Lie algebra. In those cases being in the nullcone and being nilpotent in the usual sense coincide, and the Jordan decomposition respects the grading.

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