Normality of Orthogonal and Sympletic Nilpotent Orbit Closures in Positive Characteristic

TL;DR

This paper studies when the closures of orthogonal and symplectic nilpotent orbits are normal in positive characteristic, establishing conditions related to minimal irreducible degenerations.

Contribution

It provides a criterion linking the normality of orbit closures to the absence of certain minimal irreducible degenerations in positive characteristic.

Findings

Normality of orbit closures is characterized by the absence of type d or e minimal irreducible degenerations.

The result extends a complex characteristic theorem to positive characteristic.

A complete list of minimal irreducible degenerations is used for the analysis.

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] (or see [11] from which we take Table 1 for the complete list of all minimal irreducible degenerations). 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.