Pieces of nilpotent cones for classical groups

Abstract

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