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.
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.
Accepté le :
Publié le :
@article{CRMATH_2018__356_8_852_0, author = {Le Floch, Bruno and Smilga, Ilia}, 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}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2018.06.005/} }
TY - JOUR AU - Le Floch, Bruno AU - Smilga, Ilia TI - Action of Weyl group on zero-weight space JO - Comptes Rendus. Mathématique PY - 2018 SP - 852 EP - 858 VL - 356 IS - 8 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2018.06.005/ DO - 10.1016/j.crma.2018.06.005 LA - en ID - CRMATH_2018__356_8_852_0 ER -
%0 Journal Article %A Le Floch, Bruno %A Smilga, Ilia %T Action of Weyl group on zero-weight space %J Comptes Rendus. Mathématique %D 2018 %P 852-858 %V 356 %N 8 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2018.06.005/ %R 10.1016/j.crma.2018.06.005 %G en %F CRMATH_2018__356_8_852_0
Le Floch, Bruno; Smilga, Ilia. Action of Weyl group on zero-weight space. Comptes Rendus. Mathématique, Tome 356 (2018) no. 8, pp. 852-858. doi : 10.1016/j.crma.2018.06.005. http://archive.numdam.org/articles/10.1016/j.crma.2018.06.005/
[1] On local isometric immersions of Riemannian symmetric spaces, Tohoku Math. J., Volume 36 (1984), pp. 107-140
[2] Éléments de mathématique, groupes et algèbres de Lie : chapitres 4, 5 et 6, Hermann, 1968
[3] Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Springer International Publishing, 2015
[4] Weyl group representations on zero weight spaces, 2014 http://people.math.umass.edu/~jeh/pub/zero.pdf
[5] Lie Groups Beyond an Introduction, Birkhäuser, 1996
[6] Invariant Theory, Springer, 1994
[7] Proper affine actions: a sufficient criterion (submitted, available at) | arXiv
[8] SageMath, the Sage Mathematics Software System (Version 8.1), 2017 http://www.sagemath.org
[9] Branching rules http://doc.sagemath.org/html/en/reference/combinat/sage/combinat/root_system/branching_rules.html
[10] LiE, a package for Lie group computations, 2000 http://wwwmathlabo.univ-poitiers.fr/~maavl/LiE/
Cité par Sources :