Nonuniform measure rigidity

We consider an ergodic invariant measure mu for a smooth action alpha of Z(k), k >= 2, on a (k + 1)-dimensional manifold or for a locally free smooth action of R-k, k >= 2, on a (2k +1)-dimensional manifold. We prove that if mu is hyperbolic with the Lyapunov hyperplanes in general position and if one element in Z(k) has positive entropy, then mu is absolutely continuous. The main ingredient is absolute continuity of conditional measures on Lyapunov foliations which holds for a more general class of smooth actions of higher rank abelian groups.