The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory

We make use of representation theory to study the first smooth almost-cohomology of some higher-rank abelian actions by parabolic operators. First, let N be the upper-triangular group of SL(2,C), Gamma any lattice and pi = L-2(SL(2,C)/Gamma) the usual left-regular representation. We show that the first smooth almost-cohomology group H-a(1)(N,pi) similar or equal to H-a(1)(SL(2,C),pi). In addition, we show that the first smooth almost-cohomology of actions of certain higher-rank abelian groups A acting by left translation on (SL(2,R) x G)/Gamma trivialize, where G = SL(2,R) or SL(2,C) and Gamma is any irreducible lattice. The abelian groups A are generated by various mixtures of the diagonal and/or unipotent generators on each factor. As a consequence, for these examples we prove that the only smooth time changes for these actions are the trivial ones (up to an automorphism).