  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.
