Parametrized K\"{a}hler class and Zariski dense orbital 1-cohomology

TL;DR

This paper introduces a parametrized Kähler class for Zariski dense cocycles into isometry groups of Hermitian symmetric spaces, showing it uniquely determines the cocycle up to cohomology.

Contribution

It defines a new invariant called the parametrized Kähler class for measurable cocycles and proves its completeness in classifying such cocycles up to cohomology.

Findings

The parametrized Kähler class completely determines the cocycle up to cohomology.

The notion applies to Zariski dense measurable cocycles into isometry groups of Hermitian symmetric spaces.

Provides a new tool for understanding the structure of group actions on Hermitian symmetric spaces.

Abstract

Let $\Gamma$ be a finitely generated group and let $(X,\mu_X)$ be an ergodic standard Borel probability $\Gamma$-space. Suppose that $G$ is the connected component of the identity of the isometry group of a Hermitian symmetric space. Given a Zariski dense measurable cocycle $\sigma:\Gamma \times X \rightarrow G$, we define the notion of parametrized K\"{a}hler class and we show that it completely determines the cocycle up to cohomology.x