Cocycle rigidity of abelian partially hyperbolic actions

Suppose G is a higher-rank connected semisimple Lie group with finite center and without compact factors. Let G = G or G = G ⋉ V, where V is a finite-dimensional vector space V. For any unitary representation (π,H) of G, we study the twisted cohomological equation π(a)f − λf = g for partially hyperbolic element a ∈ G and λ ∈ U(1), as well as the twisted cocycle equation π(a1)f − λ1f = π(a2)g − λ2g for commuting partially hyperbolic elements a1, a2 ∈ G. We characterize the obstructions to solving these equations, construct smooth solutions and obtain tame Sobolev estimates for the solutions. These results can be extended to partially hyperbolic flows in parallel.