Action of Weyl group on zero weight space

Abstract

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 nontrivially on the zero weight space. Among irreducible representations that have zero among their weights, $w_0$ acts by $\pm$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.

20 pages. This version has two appendixes that did not appear in the published version. The first appendix contains the full list of the results of the computations done "by hand", and the second appendix contains the computer code used to do these computations