Symplectic structure of character varieties I would like to study character varieties $\mathrm{Hom}(\Gamma,G)/G$, where $\Gamma=\langle a_1,...,a_{2g} | \Pi_{i=1}^g [a_{2i-1},a_{2i}]\rangle$ is the fundamental group of a closed surface of genus $g$ and $G$ is a connected Lie group acting by conjugation on the set of homomorphisms from $\Gamma$ to $G$. First I have to understand the motivation from the deformations of holonomies of $(G,X)$-structures.
What I already 'know': $\mathrm{Hom}(\Gamma,G)$ is a submanifold of $G^{2g}$ with possible singularities (an algebraic variety when $G$ is an algebraic group). In general, $\mathrm{Hom}(\Gamma,G)/G$ is not Hausdorff (unless $G$ is compact), nor smooth. If $G$ is reductive, we can restrict to reductive representations to get a nice quotient. See The Symplectic Nature of Fundamental Groups of Surfaces
If $G$ is compact and simply connected, $\mathrm{Hom}(\Gamma,G)$ is connected (thus $\mathrm{Hom}(\Gamma,G)/G$ too). If $G=\mathrm{SL}(2,\mathbb{R})$, with a help of Toledo invariant and a Milnor-Wood inequality, $\mathrm{Hom}(\Gamma,G)/G$ has a finite number of connected components.

How to study these topics from differential geometry view point without heavy use of GIT and algebraic groups theory. I do appreciate any hints, with textbooks or nice survey articles.

Thanks for any help!
 A: As a more modern reference, comparing to Goldman's paper,  one can use
F. Labourie, Lectures on
Representations
of Surface Groups.
EMS publishing house, pp.145,
2013, Zurich Lectures in Advanced Mathematics.
These notes provide a substantial background material on the subject.
In general, whenever $G$ is a real algebraic Lie group (pretty much any group which you will think of will fit the bill), then for every finitely-generated group $\Gamma$,
$$
Hom(\Gamma,G)
$$
has finitely many connected components. Compactness of $G$ is irrelevant. However, if you want to count these components (provided that $\Gamma$ is a surface group), or, more generally, understand their topology, then you will need some differential geometry tools (Higgs bundles). See for instance:
Peter B. Gothen, Representations of surface groups and Higgs bundles.
Wentworth, Richard A., Higgs bundles and local systems on Riemann surfaces, Álvarez-Consul, Luis (ed.) et al., Geometry and quantization of moduli spaces. Based on 4 courses, Barcelona, Spain, March – June 2012. Basel: Birkhäuser/Springer (ISBN 978-3-319-33577-3/pbk; 978-3-319-33578-0/ebook). Advanced Courses in Mathematics – CRM Barcelona, 165-219 (2016). ZBL1388.30052.
