Nilpotent pieces in the dual of odd orthogonal Lie algebras

TL;DR

This paper explores the structure of nilpotent elements in the dual of odd orthogonal Lie algebras, extending Lusztig's partitioning approach from types A, C, and D to type B.

Contribution

It provides an analysis of nilpotent pieces in the dual of odd orthogonal Lie algebras, completing Lusztig's partitioning framework for all classical types.

Findings

Partition of nilpotent elements in type B established

Explicit descriptions of nilpotent pieces provided

Extension of Lusztig's results to type B

Abstract

Let $\mathcal{N}_{\mathfrak{g}^*}$ be the variety of nilpotent elements in the dual of the Lie algebra of a reductive algebraic group over an algebraically closed field. In \cite{Lu2} Lusztig proposes a definition of a partition of $\mathcal{N}_{\mathfrak{g}^*}$ into smooth locally closed subvarieties (which are indexed by the unipotent classes in the corresponding group over complex numbers) and gives explicit results in types $A$, $C$ and $D$. We discuss type $B$ in this note.