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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

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Propriétés de maximalité concernant une représentation définie par Lusztig
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by J.-L. Waldspurger PDF
Represent. Theory 23 (2019), 379-438 Request permission

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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Additional Information
  • J.-L. Waldspurger
  • Affiliation: CNRS-Institut de Mathématiques de Jussieu-PRG, 4 place Jussieu, Boîte courrier 247, 75252 Paris cedex 05
  • MR Author ID: 180090
  • Email: jean-loup.waldspurger@imj-prg.fr
  • Received by editor(s): June 8, 2018
  • Received by editor(s) in revised form: August 22, 2019
  • Published electronically: September 30, 2019
  • © Copyright 2019 American Mathematical Society
  • Journal: Represent. Theory 23 (2019), 379-438
  • MSC (2010): Primary 05E10, 20C30
  • DOI: https://doi.org/10.1090/ert/533
  • MathSciNet review: 4013117