Action of Weyl group on zero-weight space

For any simple complex Lie group, we classify irreducible finite-dimensional representations ρ 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 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.