New examples of local rigidity of algebraic partially hyperbolic actions.

This is the first in a series of papers exploring rigidity properties of exceptional algebraic actions. We show $C^\infty$ local rigidity for a class of new examples of solvable algebraic partially hyperbolic actions on $\mathbb{G}=\mathbb{G}_1\times\cdots\times \mathbb{G}_k/\Gamma$, where $\mathbb{G}_1=SL(n,\mathbb{R})$, $n\geq3$. These examples include rank-one partially hyperbolic actions and actions enjoy minimal hyperbolicity. The method of proof is a combination of KAM type iteration scheme and representation theory. The principal difference with previous work that used KAM scheme is very general nature of the proof: no specific information about unitary representations of $\mathbb{G}$ or $\mathbb{G}_1$ is required.