Parametrized Kahler 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 X is a Hermitian symmetric space not of tube type and assume that G = Isom(X)(degrees )is simple. Given a Zariski dense measurable cocycle sigma: Gamma x X -> G,we define the notion of parametrized Kahler class and we show tha tit completely determines the cocycle up to cohomology.