Let $\mathrm{G}$ be a semi-simple algebraic group over $\mathbb{C}$ and $V$ be a finite dimensional representation of $\mathrm{G}$. Let $x \in V$ be a non zero vector such that the variety $\mathrm{G}.[x] \subset \mathbb{P}(V)$ is closed.

Is there an effective criterion to decide when the projective dual of $X$ is the closure of an orbit for the action of $\mathrm{G}$ on $\mathbb{P}(V^*)$?

I know, for instance, that if $\mathrm{G}$ acts on $\mathbb{P}(V)$ with finitely many orbits, then the projective dual of $X$ is clearly the closure of an orbit for the action of $\mathrm{G}$ on $\mathbb{P}(V^*)$.

I was wondering if there are other known cases, where this could happen. Is it the case for adjoint varieties for instance?

Thanks in advance for your help!

oneclosed orbit in $\mathbf P(V)$ (e.g. Fulton-Harris 23.52). $\endgroup$ – Francois Ziegler Jul 27 '17 at 17:16