Local rigidity of certain partially hyperbolic actions of product type

We prove certain rigidity properties of higher-rank abelian product actions of the type α× { Id}_N:{ Z}^κ(M× N), where α is (TNS) (i.e. is hyperbolic and has some special structure of its stable distributions). Together with a result about product actions of property (T) groups, this implies the local rigidity of higher-rank lattice actions of the form α× { Id}T:Γ(M), provided α has some rigidity properties itself, and contains a (TNS) subaction.