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.



Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
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](https://doi.org/10.1016/0021-8693(80)90141-6) (with some corrections in [Dadok–Kac - Polar representations](https://doi.org/10.1016/0021-8693(85)90136-X)). 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.


