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Let $\lambda$ be a symplectic partition, denote Jord^{bp}($\lambda$) the setof even positive integers i which appear in $\lambda$, and let a map$\epsilon:Jord^{bp}(\lambda) \to {\pm 1}$. The generalized Springer'scorrespondence associates to $(\lambda,\epsilon)$ an irreducible representation$\rho(\lambda,\epsilon)$ of some Weyl group. We can also define arepresentation $\underline{\rho}(\lambda,\epsilon)$ of the same Weyl group, ingeneral reducible. Roughly speaking, $\rho(\lambda,\epsilon)$ is therepresentation of the Weyl group in the top cohomology group of some varietyand $\underline{\rho}$ is the representation in the sum of all the cohomologygroups of the same variety. The representation $\underline{\rho}$ decomposes asa direct sum of $\rho(\lambda',\epsilon')$ with some multiplicities, where$(\lambda',\epsilon')$ describes the pairs similar to $(\lambda,\epsilon)$. Itis well know that $(\lambda,\epsilon)$ appears in this decomposition withmultiplicity 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 onlyeven 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$(\lambda',\epsilon')=(\lambda^{max},\epsilon^{max})$.

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