New examples of local rigidity of algebraic partially hyperbolic actions

Zhenqi Jenny Wang

Published 2019-08-24Version 1

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 $\GG=\mathbb{G}_1\times\cdots\times \mathbb{G}_k/\Gamma$, where $\mathbb{G}_1=SL(n,\RR)$, $n\geq3$. 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 $\GG$ or $\GG_1$ is required.