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Let $G$ be a connected, simply-connected complex semisimple linear algebraic group, and let $V$ be a finite-dimensional complex $G$-module. Note that if $V$ is the adjoint representation of $G$, then nilpotent $G$-orbits are invariant under the scaling action of $\mathbb{C}^*$. For arbitrary $V$, is there some kind of description of $\mathbb{C}^*$-invariant $G$-orbits? Certainly, a necessary condition is that $0$ lie in the closure of such an orbit.

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It's not clear to me whether the choice of a specific algebraically closed field matters here (though the characteristic might matter). In any case, weight vectors relative to a fixed maximal torus would be a natural starting point. And there might be some relevant results in the extensive literature on reductive group orbits in such representations. (Also, it wouldn't hurt to document the assertion in the second sentence: "Note that ...") – Jim Humphreys Oct 23 '13 at 1:06

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