Propri\'et\'es de maximalit\'e concernant une repr\'esentation d\'efinie par Lusztig

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