Nilpotent coadjoint orbits in small characteristic

Abstract

We show that the numbers of nilpotent coadjoint orbits in the dual of exceptional Lie algebra G2 in characteristic 3 and in the dual of exceptional Lie algebra F4 in characteristic 2 are finite. We determine the closure relation among nilpotent coadjoint orbits in the dual of Lie algebras of type B,C,F4 in characteristic 2 and in the dual of Lie algebra of type G2 in characteristic 3. In each case we give an explicit description of the nilpotent pieces in the dual defined in [2], which are in general unions of nilpotent coadjoint orbits, coincide with the earlier case-by-case definition in [11,26] 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.