Nilpotent Orbits in the Symplectic and Orthogonal Groups

Let K be an algebraically closed field of arbitrary characteristic and consider a linear algebraic group G over K and its Lie algebra g. For X an element of g, denote by O the orbit of X under the action of G on g defined by the adjoint representation. The Zariski closure of O is then a subvariety of g. For G equal to either the orthogonal or symplectic group, Kraft and Procesi showed that the closure of O is a normal variety for certain nilpotent elements X in g when the characteristic of K is equal to 0. We begin to generalize their result for positive, odd characteristics, concluding that the orbit closure of a nilpotent element X is normal if and only if it is normal in the union of O with all orbits of codimension two contained in the boundary of the closure of O. In particular, if the boundary of the closure of O does not contain any orbits of codimension two, then O is normal.