arXiv2017
Let $λ$ be a symplectic partition, denote Jord^{bp}($λ$) the set of even positive integers i which appear in $λ$, and let a map $ε:Jord^{bp}(λ) \to {\pm 1}$. The generalized Springer's correspondence associates to $(λ,ε)$ an irreducible representation $ρ(λ,ε)$ of some Weyl group. We can also define a representation $\underlineρ(λ,ε)$ of the same Weyl group, in general reducible. Roughly speaking, $ρ(λ,ε)$ is the representation of the Weyl group in the top cohomology group of some variety and $\underlineρ$ is the representation in the sum of all the cohomology groups of the same variety. The representation $\underlineρ$ decomposes as a direct sum of $ρ(λ',ε')$ with some multiplicities, where $(λ',ε')$ describes the pairs similar to $(λ,ε)$. It is well know that $(λ,ε)$ appears in this decomposition with multiplicity one and is minimal in this decomposition. That is, if $(λ',ε')$ appears, we have $λ'>λ$ or $(λ',ε')=(λ,ε)$. Assuming that $λ$ has only even parts, we prove that there exists also a maximal pair $(λ^{max},ε^{max})$. That is $4(λ^{max},ε^{max})$ appears with positive multiplicity (in fact one) and, if $(λ',ε')$ appears, we have $λ^{max}>λ'$ or $(λ',ε')=(λ^{max},ε^{max})$.