Measurable rigidity and disjointness for { Z}^k actions by toral automorphisms

We investigate joinings of strongly irreducible totally non-symplectic Anosov { Z}^k, k≥ 2 actions by toral automorphisms. We show that the existence of a non-trivial joining has strong implications for these actions, in particular, that the restrictions of the actions to a finite index subgroup of { Z}^k are conjugate over { Q}. We also obtain a description of the joining measures modulo the classification of zero entropy measures for the actions.