Nilpotent coadjoint orbits in small characteristic

TL;DR

This paper classifies nilpotent coadjoint orbits in the duals of certain exceptional Lie algebras over fields of small characteristic, establishing finiteness, closure relations, and explicit descriptions.

Contribution

It completes the classification of nilpotent coadjoint orbits in dual Lie algebras of reductive groups across all characteristics, including explicit descriptions and closure relations.

Findings

Finiteness of nilpotent coadjoint orbits in specific exceptional Lie algebras.

Explicit closure relations among orbits in small characteristic.

Descriptions of nilpotent pieces consistent across characteristics.

Abstract

We show that the numbers of nilpotent coadjoint orbits in the dual of exceptional Lie algebra $G_2$ in characteristic $3$ and in the dual of exceptional Lie algebra $F_4$ in characteristic $2$ are finite. We determine the closure relation among nilpotent coadjoint orbits in the dual of Lie algebras of type $B,C,F_4$ in characteristic $2$ and in the dual of Lie algebra of type $G_2$ in characteristic $3$. In each case we give an explicit description of the nilpotent pieces in the dual defined in \cite{CP}, which are in general unions of nilpotent coadjoint orbits, coincide with the earlier case-by-case definition in \cite{L4,X4} in the case of classical groups and have nice properties independent of the characteristic of the base field. This completes the classification of nilpotent coadjoint orbits in the dual of Lie algebras of reductive algebraic groups and the determination of closure relation among such orbits in all characteristic.