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

Let $C$ be a unipotent class of $G=\textrm{SO}(N,\mathbb{C})$, $\mathcal{E}$ an irreducible $G$-equivariant local system on $C$. The generalized Springer representation $\rho (C,\mathcal{E})$ appears in the top cohomology of some variety. Let $\bar \rho (C,\mathcal{E})$ be the representation obtained by summing over all cohomology groups of this variety. It is well known that $\rho (C,\mathcal{E})$ appears in $\bar \rho (C,\mathcal{E})$ 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 $\bar \rho (C,\mathcal{E})$. Suppose $C$ is parametrized by an orthogonal partition with only odd parts. We prove that $\bar \rho (C,\mathcal{E})$ (resp. $\textrm{sgn}\otimes \bar \rho (C,\mathcal{E})$) has a unique multiplicity 1 “maximal” subrepresentation $\rho ^{\textrm{max}}$ (resp. “minimal” subrepresentation $\textrm{sgn}\otimes \rho ^{\textrm{max}}$), where $\textrm{sgn}$ is the sign representation. These are analogues of results for $\textrm{Sp}(2n,\mathbb{C})$ by Waldspurger.