Parametrized K{a}hler class and Zariski dense Eilenberg-MacLane 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 $sigma:Gammatimes Xrightarrow G$, we define the notion of parametrized K{a}hler class and we show that it completely determines the cocycle up to cohomology.