On the action of the restricted Weyl group on the set of orbits a minimal parabolic subgroup

We construct the action of the restricted Weyl group on the set of principal families of orbits of a minimal parabolic subgroup over an algebraically nonclosed field. Additionally, we relate this action to the action on a polarized cotangent bundle. These results generalize the corresponding results of Knop on the action of the Weyl group on the families of Borel orbits of maximal complexity and rank.