Given a connected semisimple Lie group G, Monod (Trans. Amer. Math. Soc. B 144–159, 2022) has recently proved that the measurable cohomology of the G-action \(H^*_m(G \curvearrowright G/P)\) on the Furstenberg boundary G/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/L is compact. More precisely, we show that the cohomology of the G-action \(H^p_m(G \curvearrowright G/L)\) maps surjectively onto \(H^p_m(G)\) with a kernel isomorphic to \(H^{p-1}_m(L)\) . 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 \(\textrm{SL}(2,\mathbb {K})\) for \(\mathbb {K}=\mathbb {R}, \mathbb {C}\) .
