Nilpotent pieces in the dual of odd orthogonal Lie algebras

Let ${\mathcal{N}}_{{\mathfrak{g}}^{\ast }}$ 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 ${\mathcal{N}}_{{\mathfrak{g}}^{\ast }}$ 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.