*Accepted Paper*

**Inserted:** 29 jun 2021

**Last Updated:** 24 may 2023

**Journal:** to appear in Math. Res. Lett.

**Year:** 2023

**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.