Séminaires : Séminaire Groupes Réductifs et Formes Automorphes

Equipe(s) : fa, tn,
Responsables :Alexis Bouthier, Cong Xue
Email des responsables : alexis.bouthier@imj-prg.fr, cong.xue@imj-prg.fr
Salle :
Adresse :

Orateur(s) Ruben La - University of Oxford,
Titre Maximality properties of generalised Springer representations of $\text{SO}(N,\mathbb{C})$
Horaire10:30 à 12:00

Waldspurger proved maximality and minimality results for certain generalised Springer representations of $\text{Sp}(2n,\mathbb{C})$. We will discuss analogous results for $G = \text{SO}(N,\mathbb{C})$ and sketch their proofs.

Let $C$ be a unipotent class of $G$ and $E$ an irreducible $G$-equivariant local system on $C$. The generalised Springer correspondence attaches to $(C,E)$ an irreducible representation $\rho$ of a relative Weyl group of $G$. We call $C$ the support of $\rho$. It is well-known that $\rho$ appears in the top cohomology of a certain variety. Let $\bar\rho$ be the representation obtained by summing the cohomology groups of this variety.

We show that if $C$ is parametrised by an orthogonal partition consisting of only odd parts, then $\bar\rho$ has a unique irreducible subrepresentation $\rho^{\text{max}}$ whose support is maximal among the supports of the irreducible subrepresentations of $\rho^{\text{max}}$. We also show that $\text{sgn}\otimes\rho^{\text{max}}$ is the unique subrepresentation of $\text{sgn}\otimes\bar\rho$ with minimal support. We will also present an algorithm to compute $\rho^{\text{max}}$.

Salle15-25 502