TITLE: Interesting properties of "coadjoint" orbits inside $V\in \operatorname{Rep}G$
QUESTION [5 upvotes]: Let $G$ be a reductive group over $\mathbf{C}$. It acts on the dual of its Lie algebra $\mathfrak{g}^*$ by  conjugation.

One can describe the orbits of $\mathfrak{g}^*$ explicitly (e.g. using Jordan blocks for $\operatorname{GL}_n$),
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.

REPLY [5 votes]: 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.