Let $C$ be a connected compact oriented real surface of genus $g$, let $G$ be a connected compact Lie group and let $G_\mathbb{C}$ be the complexification of $G$. One considers the moduli space $M (C,G)$ of isomorphism classes of representations of the fundamental group $\pi_1 (C)$ in $G$. It is a compact real algebraic variety (in general singular) with a natural symplectic structure (see Atiyah-Bott, Goldman,...)

If one fixes a complex structure on $C$, one can consider the moduli space $M_{hol} (C,G)$ of degree zero semistable holomorphic $G_\mathbb{C}$-bundles on $C$. It is a projective complex algebraic variety (in general singular) with a natural polarization. The theorem of Narasimhan-Seshadri (for $G=U(n)$, extended by Ramanathan for general $G$) gives an isomorphism between the underlying real analytic structures of $M(C,G)$ and $M_{hol}(C,G)$, with identification of the natural symplectic form on $M(C,G)$ with the natural Kähler form on $M_{hol}(C,G)$. In other words, a choice of complex structure on $C$ gives a natural polarized complex algebraic structure on $M(C,G)$ compatible with the symplectic structure. In vague terms, my question is: how does this structure changes when the complex structure on $C$ changes?

More precisely, I would like to consider the moduli space $\mathcal{M}$ of polarized complex algebraic structures on $M(C,G)$ compatible with the symplectic structure. I don't know if such moduli space exists.

Q1) Is there a way to make sense of the moduli space $\mathcal{M}$ ? Locally, globally?

If $\mathcal{M}$ makes sense then the Narasimhan-Seshadri theorem gives a map $f: M_g \rightarrow \mathcal{M}$, $C \mapsto M_{hol}(C, G)$, where $M_g$ is the moduli space of complex structures on $C$.

Q2) What can be said on the map $f$ ? Is it injective ? What is the dimension of its image?...

When $G=U(1)$, we simply have $\mathcal{M} = A_g$, the moduli space of principally polarized abelian varieties of dimension $g$ and $f: M_g \rightarrow A_g$ is simply the Torelli map $C \mapsto Jac(C)$, which is injective by the Torelli theorem. So my questions could be reformulated as: is there some non-abelian version of the Torelli map and of the Torelli theorem (where "non-abelian" refers to the replacement of $G=U(1)$ by a general $G$) ?