For any simple complex Lie group, we classify irreducible finite-dimensional representations ρ for which the longest element of the Weyl group acts non-trivially on the zero-weight space. Among irreducible representations that have zero among their weights, acts by ±Id if and only if the highest weight of ρ is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.
Pour tout groupe de Lie complexe simple, nous classifions les représentations irréductibles ρ de dimension finie telles que le plus long mot du groupe de Weyl agisse non trivialement sur l'espace de poids nul. Parmi les représentations irréductibles dont zéro est un poids, agit par ±Id si et seulement si le plus haut poids de ρ est un multiple d'un poids fondamental, avec un coefficient plus petit qu'une borne qui dépend du groupe et du poids fondamental.
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Bruno Le Floch 1; Ilia Smilga 2
@article{CRMATH_2018__356_8_852_0, author = {Bruno Le Floch and Ilia Smilga}, title = {Action of {Weyl} group on zero-weight space}, journal = {Comptes Rendus. Math\'ematique}, pages = {852--858}, publisher = {Elsevier}, volume = {356}, number = {8}, year = {2018}, doi = {10.1016/j.crma.2018.06.005}, language = {en}, }
Bruno Le Floch; Ilia Smilga. Action of Weyl group on zero-weight space. Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 852-858. doi : 10.1016/j.crma.2018.06.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.06.005/
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