On Shimura's isomorphism and $(\Gamma, G)$-bundles on the upper-half plane

For a compact real form $U$ of a complex simple Lie group $G$, and an irreducible representation $\rho:\Gamma \to U$ of a Fuchsian group of the first kind $\Gamma$, it is shown that the classical isomorphism of Shimura, for the periods of a cusp form of weight 2 with values in $\mathfrak{g}$ and the representation $\textrm{Ad}\rho:\Gamma\to\textrm{Aut}\mathfrak{g}$, can be interpreted as the differential at a point of the zero section, for a natural map from the cotangent bundle of the moduli space of certain $(\Gamma, G)$-bundles over $\mathbb{H}$ (in the sense of Seshadri) to an open set in the smooth locus of the character variety $\textrm{Hom}_{\mathbf{t}}(\Gamma,G)/PG$. Emphasis is put on analytic techniques.