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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!

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  • $\begingroup$ Presumably $X=\mathrm{G}.[x]$, but you don't say? Note that for $V$ irreducible (as F. Knop says one can restrict attention to) there is just one closed orbit in $\mathbf P(V)$ (e.g. Fulton-Harris 23.52). $\endgroup$ – Francois Ziegler Jul 27 '17 at 17:16
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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.

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