Pieces of nilpotent cones for classical groups

We compare orbits in the nilpotent cone of type $B_n$, that of type $C_n$, and Kato’s exotic nilpotent cone. We prove that the number of ${\text{F}}_q$-points in each nilpotent orbit of type $B_n$ or $C_n$ equals that in a corresponding union of orbits, called a type-$B$ or type-$C$ piece, in the exotic nilpotent cone. This is a finer version of Lusztig’s result that corresponding special pieces in types $B_n$ and $C_n$ have the same number of ${\text{F}}_q$-points. The proof requires studying the case of characteristic 2, where more direct connections between the three nilpotent cones can be established. We also prove that the type-$B$ and type-$C$ pieces of the exotic nilpotent cone are smooth in any characteristic.