Local rigidity of higher rank partially hyperbolic algebraic actions

TL;DR

This paper establishes the $C^ abla$ local rigidity of higher rank partially hyperbolic algebraic actions on various spaces, using a novel combination of harmonic analysis, representation theory, and KAM methods, without relying on specific representation data.

Contribution

It provides the first local rigidity results for non-accessible partially hyperbolic actions beyond torus examples, including twisted symmetric spaces and automorphisms on nilmanifolds.

Findings

Proves $C^ abla$ local rigidity for abelian ergodic algebraic actions.

Extends local rigidity results to twisted symmetric space examples.

Introduces a new method combining harmonic analysis and KAM iteration without specific representation theory data.

Abstract

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.