Some explicit cocycles on the Furstenberg boundary for products of isometries of hyperbolic spaces and $\mathrm{SL}(3,\mathbb{K})$

Abstract: 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 kernel that can be entirely described in terms of invariants in the cohomology of the maximal split torus $A<G$. In a paper the authors refine Monod's result and show in particular that the cohomology of non-alternating cocycles on $G/P$, namely those lying in the kernel of the alternation map, is in general not trivial and lies in the kernel of the evaluation. In this paper we describe explicitly such non-alternating and alternating cocycles on $G/P$ in low degrees when $G$ is either a product of isometries of real hyperbolic spaces or $G=\mathrm{SL}(3,\mathbb{K})$, where $\mathbb{K}$ is either the real or the complex field. As a consequence, we deduce that the comparison map $H^*_{m,b}(G)\rightarrow H^*_m(G)$ from the measurable bounded cohomology is injective in degree $3$, which is new for nontrivial products of isometries of hyperbolic spaces.