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Propriétés de maximalité concernant une représentation définie par Lusztig

Author: J.-L. Waldspurger
Journal: Represent. Theory 23 (2019), 379-438
MSC (2010): Primary 05E10, 20C30
Published electronically: September 30, 2019
MathSciNet review: 4013117
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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})$.

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J.-L. Waldspurger
Affiliation: CNRS-Institut de Mathématiques de Jussieu-PRG, 4 place Jussieu, Boîte courrier 247, 75252 Paris cedex 05

Received by editor(s): June 8, 2018
Received by editor(s) in revised form: August 22, 2019
Published electronically: September 30, 2019
Article copyright: © Copyright 2019 American Mathematical Society