Higher secant varieties of the minimal adjoint orbit

The adjoint group of a simple complex Lie algebra g has a unique minimal orbit in the projective space Pg, whose pre-image in g we denote by C. We explicitly describe, for every classical g and every natural number k, the Zariski closure of the union kC of all spaces spanned by k points on C. The image of in Pg is usually called the (k−1)st secant variety of PC. These higher secant varieties are known, and easily determined, for g=sln or g=sp2n; for completeness, we give short proofs of these results. Our main contribution is therefore the explicit description of for g=on, where the embedding of PC into Pon is isomorphic to the Plücker embedding of the Grassmannian of isotropic lines in Pn−1 into . We show that the first and the second secant variety are then characterised by certain conditions on the eigenvalues of matrices in on, while the third and higher secant varieties coincide with those of the Grassmannian of all projective lines. Finally, unlike for g=sln or sp2n, the sets kC are not all closed in on, and we present a partial result on the nilpotent orbits contained in them.