Rigidity of higher rank abelian cocycles with values in diffeomorphism groups

We consider cocycles over certain hyperbolic R-k actions, k >= 2, and show rigidity properties for cocycles with values in a Lie group or a diffeomorphism group, which are close to identity on a set of generators, and are sufficiently smooth. The actions we consider are Cartan actions of SL(n, R)/ Gamma or SL(n, C)/ Gamma, for n >= 3, and Gamma torsion free cocompact lattice. The results in this paper rely on a technique developed recently by D. Damjanovic and A. Katok.