Lie algebras
Action of Weyl group on zero-weight space
Comptes Rendus. Mathématique, Volume 356 (2018) no. 8, pp. 852-858.

For any simple complex Lie group, we classify irreducible finite-dimensional representations ρ for which the longest element w0 of the Weyl group acts non-trivially on the zero-weight space. Among irreducible representations that have zero among their weights, w0 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 w0 du groupe de Weyl agisse non trivialement sur l'espace de poids nul. Parmi les représentations irréductibles dont zéro est un poids, w0 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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Published online:
DOI: 10.1016/j.crma.2018.06.005
Le Floch, Bruno 1; Smilga, Ilia 2

1 Princeton Center for Theoretical Science, Princeton, NJ 08544, USA
2 Yale University Mathematics Department, PO Box 208283, New Haven, CT 06520-8283, USA
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Le Floch, Bruno; Smilga, Ilia. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.06.005/

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