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

TL;DR

This paper investigates the maximality properties of certain representations arising from the generalized Springer correspondence for special orthogonal groups, establishing unique maximal and minimal subrepresentations under specific conditions.

Contribution

It proves the existence and uniqueness of a maximal subrepresentation in the Springer representation for SO(N,C) when parametrized by odd parts, extending Waldspurger's results.

Findings

Existence of a unique maximal subrepresentation with multiplicity 1.

Identification of the minimal subrepresentation after tensoring with the sign representation.

Extension of maximality/minimality results to SO(N,C) from previous work on Sp(2n,C).

Abstract

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.