Propriétés de maximalité concernant une représentation définie par Lusztig
Résumé
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 (\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}).
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