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 ρ (C,E) appears in the top cohomology of some variety. Let ρ (C,E) be the representation obtained by summing over all cohomology groups of this variety. It is well known that ρ (C,E) appears in ρ (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 ρ (C,E). Suppose C is parametrized by an orthogonal partition with only odd parts. We prove that ρ (C,E) (resp. sgn⊗ ρ (C,E)) has a unique multiplicity 1 “maximal” subrepresentation ρ ᵐᵃˣ (resp. “minimal” subrepresentation sgn⊗ ρ ᵐᵃˣ), where sgn is the sign representation. These are analogues of results for Sp(2n,C) by Waldspurger.