Local rigidity of certain partially hyperbolic actions of product type

We prove certain rigidity properties of higher-rank abelian product actions of the type alpha x Id(N) : Z(K) --> Diff(M x N), where alpha 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 alpha x Id(T) : Gamma --> Diff(M x T), provided alpha has some rigidity properties itself, and contains a (TNS) subaction.