Parametrized Kähler 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 $\mathcal{X}$ is a Hermitian symmetric space not of tube type and assume that $G=\operatorname{Isom}(\mathcal{X})^{\circ}$ is simple. Given a Zariski dense measurable cocycle $\sigma:\Gamma \times X \to G$, we define the notion of parametrized Kähler class and we show that it completely determines the cocycle up to cohomology.