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

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