Nilpotent orbits in classical Lie algebras over finite fields of characteristic 2 and the Springer correspondence

TL;DR

This paper constructs a Springer correspondence for classical Lie algebras over algebraically closed fields of characteristic 2, analyzing nilpotent orbits and component groups, and determining their counts over finite fields.

Contribution

It develops a Springer correspondence in characteristic 2 for types B, C, D, and characterizes nilpotent orbit structures and counts over finite fields.

Findings

Constructed Springer correspondence for types B, C, D in characteristic 2.

Determined structure of component groups of nilpotent centralizers.

Counted nilpotent orbits over finite fields.

Abstract

Let $G$ be an adjoint algebraic group of type $B$, $C$, or $D$ over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of $G$. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over finite fields is obtained.