  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.
