Action of Weyl group on zero weight space

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.