The secant varieties of nilpotent orbits

Let g be a complex simple Lie algebra. We have the adjoint representation of the adjoint group G on g . Then G acts on the projective space P g . We consider the closure X of the image of a nilpotent orbit in P g . The i -secant variety Sec ( i ) X of a projective variety X is the closure of the union of projective subspaces of dimension i in the ambient space P spanned by i + 1 points on X . In particular we call the 1-secant variety the secant variety. In this paper we give explicit descriptions of the secant and the higher secant varieties of nilpotent orbits of complex classical simple Lie algebras.