 
Summary: PIECES OF NILPOTENT CONES FOR CLASSICAL GROUPS
PRAMOD N. ACHAR, ANTHONY HENDERSON, AND ERIC SOMMERS
Abstract. We compare orbits in the nilpotent cone of type Bn, that of type
Cn, and Kato's exotic nilpotent cone. We prove that the number of Fqpoints
in each nilpotent orbit of type Bn or Cn equals that in a corresponding union
of orbits, called a typeB or typeC piece, in the exotic nilpotent cone. This
is a finer version of Lusztig's result that corresponding special pieces in types
Bn and Cn have the same number of Fqpoints. 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 typeB and typeC
pieces of the exotic nilpotent cone are smooth in any characteristic.
1. Introduction
Let F be an algebraically closed field; for the moment, assume that the charac
teristic of F is not 2. The algebraic groups SO2n+1(F) (of type Bn) and Sp2n(F)
(of type Cn) share many features by virtue of having dual root data and hence
isomorphic Weyl groups: W(Bn) = W(Cn) = {±1} Sn.
The connection between the orbits in their respective nilpotent cones N(o2n+1)
and N(sp2n) is subtle. (Recall that in characteristic = 2, these nilpotent cones are
isomorphic to the unipotent varieties of the corresponding groups, so everything we
say about nilpotent orbits implies an analogous statement for unipotent classes.)
