Kernels in measurable cohomology for transitive actions

TL;DR

This paper extends Monod's results on measurable cohomology of Lie group actions, showing similar surjectivity and kernel descriptions for actions on quotients by certain subgroups, with explicit cocycle computations for SL(2,K).

Contribution

It generalizes Monod's theorem to actions on quotients by subgroups with compact stabilizers, identifying the kernel as cohomology of the subgroup, and provides explicit cocycle examples.

Findings

Surjective maps from measurable cohomology of G-actions to G's cohomology.

Kernel of the cohomology map is isomorphic to the cohomology of the subgroup L.

Explicit cocycle computations for SL(2,R) and SL(2,C).

Abstract

Given a connected semisimple Lie group $G$, Monod 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 $\mathrm{SL}(2,\mathbb{K})$ for $\mathbb{K}=\mathbb{R}, \mathbb{C}$.