Damjanovic and A. Katok proved, using the KAM method, that partially hyperbolic higher rank abelian actions by ergodic automorphisms on the torus are locally rigid. I will discuss the corresponding result in the more general case of nilmanifold automorphisms. Using the representation theory of 2-step free nilpotent Lie groups, we obtain tame estimates for the solution of the linearized problem - the first step in the KAM scheme. Reduction to the linearized problem by tame estimates is in progress.

This is joint work with Damjanovic and Török.