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!



dense orbit projective dual homogeneous space
This is certainly a very rare phenomenon. First of all, since $Gx$ generates an irreducible submodule of $V$ you may restrict your attention the case that $V$ is irreducible.


Next, the dual variety of $X$ is almost always of codimension one. The few exceptions were classified in Knop-Menzel (1987).


Now assume we are in the adjoint case $V=\text{Lie}(G)$ where $G$ is simple. Then, $\mathbb P(V^\*)$ contains an orbit of codimension $1$, hence $V^\*=V$ contains an orbit of codimension $\le2$. This forces $\text{rk}(G)\le2$. Conversely, this condition is also easily seen to be sufficient (in fact one shows that every irreducible divisor is an orbit closure).


So, in the adjoint case one gets $G=SL(2), SL(3), Sp(4)$, and $\mathsf G\_2$.


Remark: In general, since $V$ contains an orbit of codimension $\le2$ also the generic orbit is of codimension $\le2$ forcing $\dim V/\!/G\le 2$. Then $V/\!/G$ is smooth by Kempf (1980). But irreducible representations with smooth quotient have been classified in Littelmann (1989). I am confident that most $V$ with $V/\!/G\le2$ provide examples. 


