Kernels in measurable cohomology for transitive actions

Given a connected semisimple Lie group $G$, Monod has recently proved that the measurable cohomology of the $G$-action $H_m^{\ast }(G\curvearrowright G\mathrm{/}P)$ on the Furstenberg boundary $G\mathrm{/}P$, where $P$ is a minimal parabolic subgroup, maps surjectively on the measurable cohomology of $G$ through the evaluation on a fixed basepoint. Additionally, the kernel of this map depends entirely on the invariant cohomology of a maximal split torus. In this paper we show a similar result for a fixed subgroup $L<P$ such that the stabilizer of almost every pair of points in $G\mathrm{/}L$ is compact. More precisely, we show that the cohomology of the $G$-action $H_m^p(G\curvearrowright G\mathrm{/}L)$ maps surjectively onto $H_m^p(G)$ with a kernel isomorphic to $H_m^{p-1}(L)$. Examples of such groups are given either by any term of the derived series of the unipotent radical $N$ of $P$ or by a maximal split torus $A$. We conclude the paper by computing explicitly some cocycles on quotients of $\mathrm{S}\mathrm{L}(2,\mathbb{K})$ for $\mathbb{K}=\mathbb{R},\mathbb{C}$.