Propriétés de maximalité concernant une représentation définie par Lusztig

Abstract:

Let $\lambda$ be a symplectic partition, denote $Jord^{bp}(\lambda )$ the set of even positive integers 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 $\underline {\rho }(\lambda ,\epsilon )$ 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 set of 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, $(\lambda ^{max},\epsilon ^{max})$ appears with positive multiplicity (in fact one) and, if $(\lambda ’,\epsilon ’)$ appears, we have $\lambda ^{max}>\lambda ’$ or $(\lambda ’,\epsilon ’)=(\lambda ^{max},\epsilon ^{max})$.