Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra--one nilpotent, and the other semisimple. Second, we consider actions from two commuting unipotent flows coming from two commuting embedded copies of SL(2,R). In both cases we show that any smooth real-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function.