Normality of Orthogonal and Sympletic Nilpotent Orbit Closures in Positive Characteristic

Abstract: In this note we investigate the normality of closures of orthogonal and symplectic nilpotent orbits in positive characteristic. We prove that the closure of such a nilpotent orbit is normal provided that neither type d nor type e minimal irreducible degeneration occurs in the closure, and conversely if the closure is normal, then any type e minimal irreducible degeneration does not occur in it. Here, the minimal irreducible degenerations of a nilpotent orbit are introduced by W. Hesselink in 6. Our result is a weak version in positive characteristic of [11, Theorem 16.2(ii)], one of the main results of [11] over complex numbers.