Alternating cochains on Furstenberg boundaries and measurable cohomology

Nicolas Monod showed that the evaluation map $$H^*_m(G\curvearrowright G/P)\longrightarrow H^*_m(G)$$ between the measurable cohomology of the action of a connected semisimple Lie group $G$ on its Furstenberg boundary $G/P$ and the measurable cohomology of $G$ is surjective with a non-trivial kernel in all degrees below a constant depending on $G$ and less than or equal to the rank of $G$ plus $2$. When we were looking for explicit representatives of classes in this kernel, we were astonished to discover that some of these nontrivial classes have trivial alternation. In this paper, we refine Monod's result by identifying the non-alternating and alternating cohomology classes in this kernel. As a consequence, we show that $H^*_m(G)$ is isomorphic to the alternating measurable cohomology of $G$ acting on $G/P$ in all even degrees $$H^{2k}_{m,\mathrm{alt}}(G\curvearrowright G/P)\cong H^{2k}_m(G),$$ for a majority of Lie groups, namely those for which the longest element of the Weyl group acts as $-1$ on the Lie algebra of a maximal split torus $A$ in $G$.