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!