  We prove absolute continuity of "high entropy" hyperbolic invariant measures
for smooth actions of higher rank abelian groups assuming that there are no
proportional Lyapunov exponents. For actions on tori and infranilmanifolds
existence of an absolutely continuous invariant measure of this kind is
obtained for actions whose elements are homotopic to those of an action by
hyperbolic automorphisms with no multiple or proportional Lyapunov exponents.
In the latter case a form of rigidity is proved for certain natural classes of
cocycles over the action.
