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

Ruben La
Aug 2022
摘要
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 $C$, anirreducible 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 topcohomology of some variety. Let $\bar\rho(C,\mathcal{E})$ be the representationobtained by summing over all cohomology groups of this variety. It iswell-known that $\rho(C,\mathcal{E})$ appears in $\bar\rho(C,\mathcal{E})$ withmultiplicity $1$ and that it is a `minimal subrepresentation' in the sense thatits Springer support $C$ is strictly minimal in the closure ordering among theSpringer supports of the irreducbile subrepresentations of$\bar\rho(C,\mathcal{E})$. Suppose $C$ is parametrised by an orthogonalpartition 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 thesign representation of the relevant relative Weyl group. We also show that$\mathrm{sgn} \otimes \rho(C^{\mathrm{max}},\mathcal{E}^{\mathrm{max}})$ is theminimal subrepresentation of $\mathrm{sgn} \otimes \bar\rho(C,\mathcal{E})$.These results are direct analogues of similar maximality and minimality resultsfor $\mathrm{Sp}(2n,\mathbb{C})$ by Waldspurger.
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