The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory (Q874770)

From the introduction: ``In this article, we will compute the first smooth (or \(C^r\)) almost-cohomology \(H^1(A, L^2(G/\Gamma))\) for several examples of higher-rank parabolic abelian actions, and show that in some sense, the cohomology is as small as possible. In particular (with one notable exception) in each of our examples we show, for all irreducible unitary representations \(\pi\) of \(G\), that \(H^1_a(A, \pi)=0\). That is, all nontrivial obstructions to the cohomological equations vanish, and the first smooth (or \(C^r\)) almost-cohomology trivializes. Our method also works for some hyperbolic and mixed actions.''