Waldspurger proved maximality and minimality results for certain generalised Springer representations of $\text{Sp}(2n,\mathbb{C})$. In this talk, we will discuss analogous results for $G = \text{SO}(N,\mathbb{C})$ and we will give a sketch of their proofs. The proofs are entirely combinatorial.

The setting is as follows. The generalised Springer correspondence for $G$ attaches to a pair $(C,E)$, where $C$ is a unipotent class of $G$ and E is an irreducible $G$-equivariant local system on $C$, an irreducible representation $\rho(C,E)$ of a relative Weyl group of $\text{SO}(N)$. We call $C$ the Springer support of $\rho(C,E)$. Each $\rho(C,E)$ appears in the top cohomology of a certain variety. Let $\bar\rho(C,E)$ be the representation obtained by summing the cohomology groups of this variety. Suppose $C$ is parametrised by an orthogonal partition consisting of only odd parts.

We show that there exists a unique pair $(C^{\text{max}},E^{\text{max}})$ such that $\rho(C^{\text{max}},E^{\text{max}})$ appears with multiplicity 1 in $\bar\rho(C,E)$ and $C^{\text{max}}$ is strictly maximal among the Springer supports of the irreducible subrepresentations of $\bar\rho(C,E)$. Let $(C^{\text{min}},E^{\text{min}})$ be such that $\rho(C^{\text{min}},E^{\text{min}}) = \text{sgn}\otimes\rho(C^{\text{max}},E^{\text{max}})$. We show that $\rho(C^{\text{min}},E^{\text{min}})$ appears with multiplicity 1 in $\text{sgn}\otimes\bar\rho(C,E)$ and $C^{\text{min}}$ is strictly minimal among the Springer supports of the irreducible constituents of $\text{sgn}\otimes\bar\rho(C,E)$. We will also present an algorithm to compute $(C^{\text{max}},E^{\text{max}})$.

Seminar series

Date

Fri, 07 Oct 2022

Time

12:00 -
13:00

Location

C3

Speaker

Ruben La

Organisation

University of Oxford