Alternating cochains on Furstenberg boundaries and measurable cohomology

TL;DR

This paper refines Monod's results on measurable cohomology of semisimple Lie groups by distinguishing between alternating and non-alternating classes, showing isomorphisms in even degrees for many groups.

Contribution

It identifies the alternating and non-alternating parts of the kernel in Monod's surjectivity result, establishing isomorphisms with alternating cohomology in even degrees for certain Lie groups.

Findings

Nontrivial classes with trivial alternation discovered.

Isomorphism between $H^{2k}_m(G)$ and alternating cohomology established.

Results hold for Lie groups where the Weyl group's longest element acts as -1.

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 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 $G$.