Let $G$ be a connected, simplyconnected complex semisimple linear algebraic group, and let $V$ be a finitedimensional 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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