Higher secant varieties of the minimal adjoint orbit
The adjoint group of a simple complex Lie algebra $\liea{g}$ has a
unique minimal orbit in the projective space $\PP\liea{g}$, whose
pre-image in $\liea{g}$ we denote by $C$. We explicitly describe, for
every classical $\liea{g}$ and every natural number $k$, the Zariski
closure $\overline{kC}$ of the union $kC$ of all spaces spanned by $k$
points on $C$. The image of $\overline{kC}$ in $\PP \liea{g}$ is usually
called the $(k-1)$-st {\em secant variety} of $\PP C$. These higher secant
varieties are known, and easily determined, for $\liea{g}=\liea{sl}_n$
or $\liea{g}=\liea{sp}_{2n}$; for completeness, we give short proofs of
these results. Our main contribution is therefore the explicit description
of $\overline{kC}$ for $\liea{g}=\liea{o}_n$, where the embedding of $\PP
C$ into $\PP \liea{o}_n$ is isomorphic to the Pl\"ucker embedding of the
Grassmannian of isotropic lines in $\PP^{n-1}$ into
$\PP^{\frac{(n+1)n}{2}-1}$. We show that the first and the second
secant variety are then characterised by certain conditions on the
eigenvalues of matrices in $\liea{o}_n$, while the third and higher secant
varieties coincide with those of the Grassmannian of {\em all} projective
lines. Finally, unlike for $\liea{g}=\liea{sl}_n$ or $\liea{sp}_{2n}$,
the sets $kC$ are not all closed in $\liea{o}_n$, and we present a
partial result on the nilpotent orbits contained in them.
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