Maximality Properties of Generalized Springer Representations of SO(N, C)

Let C be a unipotent class of G = SO(N, C), E an irreducible G-equivariant local system on C. The generalized Springer representation rho(C, E) appears in the top cohomology of some variety. Let (rho) over bar (C, E) be the representation obtained by summing overall cohomology groups of this variety. It is well known that rho(C, E) appears in (rho) over bar (C, E) with multiplicity 1 and that its Springer support C is strictly minimal in the closure ordering among the Springer supports of the irreducbile subrepresentations of (rho) over bar (C, E). Suppose C is parametrized by an orthogonal partition with only odd parts. We prove that (rho) over bar (C, E) (resp. sgn circle times (rho) over bar (C, E)) has a unique multiplicity 1 "maximal" subrepresentation rho(max)(resp."minimal" subrepresentation sgn circle times rho(max)), where sgn is the sign representation. These are analogues of results for Sp(2n, C) by Waldspurger.