Maximality properties of generalised Springer representations of $\mathrm{SO}(N,\mathbb{C})$

The generalised Springer correspondence for $G = \mathrm{SO}(N,\mathbb{C})$ attaches to a pair $(C,\mathcal{E})$, where $C$ is a unipotent class of $G$ and $\mathcal{E}$ is an irreducible $G$-equivariant local system on $C$, an irreducible representation $\rho(C,\mathcal{E})$ of a relative Weyl group of $G$. We call $C$ the Springer support of $\rho(C,\mathcal{E})$. For each such $(C,\mathcal{E})$, $\rho(C,\mathcal{E})$ appears with multiplicity 1 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 it is a `minimal subrepresentation' in the sense 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 parametrised by an orthogonal partition consisting of only odd parts. We prove that there exists a unique `maximal subrepresentation' $\rho(C^{\mathrm{max}},\mathcal{E}^{\mathrm{max}})$ of multiplicity $1$ of $\bar\rho(C,\mathcal{E})$. Let $\mathrm{sgn}$ be the sign representation of the relevant relative Weyl group. We also show that $\mathrm{sgn} \otimes \rho(C^{\mathrm{max}},\mathcal{E}^{\mathrm{max}})$ is the minimal subrepresentation of $\mathrm{sgn} \otimes \bar\rho(C,\mathcal{E})$. These results are direct analogues of similar maximality and minimality results for $\mathrm{Sp}(2n,\mathbb{C})$ by Waldspurger.