  We give a complete solution to the local classification program of higher
rank partially hyperbolic algebraic actions. We show $C^\infty$ local rigidity
of abelian ergodic algebraic actions for symmetric space examples, twisted
symmetric space examples and automorphisms on nilmanifolds. The method is a
combination of representation theory, harmonic analysis and a KAM iteration. A
striking feature of the method is no specific information from representation
theory is needed. It is the first time local rigidity for non-accessible
partially hyperbolic actions has ever been obtained other than torus examples.
Even for Anosov actions, our results are new: it is the first time twisted
spaces with non-abelian nilradical have been treated in the literature.
