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## Action of Weyl group on zero-weight space

Comptes Rendus Mathematique. 2018. Vol. 356. No. 8 . P. 852-858.
Le Floch B., Smilga I.

For any simple complex Lie group, we classify irreducible finite-dimensional representations rho for which the longest element w_0 of the Weyl group acts non-trivially on the zero-weight space. Among irreducible representations that have zero among their weights, w_0 acts by Id or -Id if and only if the highest weight of rho is a multiple of a fundamental weight, with a coefficient less than a bound that depends on the group and on the fundamental weight.