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 $SL(2,\mathbb{C})$, $\Gamma$ any lattice and $\pi = L^2(SL(2,\mathbb{C})$/$\Gamma)$ the usual left-regular representation. We show that the first smooth almost-cohomology group $H_a^1(N, \pi)$ ≃ $H_a^1(SL(2,\mathbb{C}) , \pi)$. In addition, we show that the first smooth almost-cohomology of actions of certain higher-rank abelian groups $A$ acting by left translation on $(SL(2,\mathbb{R}) \times G)$/$\Gamma$ trivialize, where $G = SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$ 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).