Let $\Sigma$ be a Riemann surface of genus $g$. We can consider two different type of objects associated to it parametrising representations of its fundamental groups. On one side we have the character stack $$\mathcal{X}=[\{A_i,B_i \in GL(n,\mathbb{C}) \ | \ \prod_{i=1}^g[A_i,B_i]=1 \}/ Gl(n,\mathbb{C})]$$ and on the other side we have the GIT quotient $$X=\{A_i,B_i \in GL(n,\mathbb{C}) \ | \ \prod_{i=1}^g[A_i,B_i]=1 \}//Gl(n,\mathbb{C}) .$$
From general theory, we have a morphism $f:\mathcal{X} \to X$. I was wondering what is known about $\mathcal{X}$ and its relation with $X$ in particular referring to their cohomologies. I've crossed into an article https://arxiv.org/abs/1504.00352 which relates the E-polynomials of $\mathcal{X}$ to that of the twisted smooth versions of $X$.
This may be a very vague question but I was also wondering which of the two objects should be considered more "interesting" from a mathematical viewpoint. A possible interpretation of this should be the object with the richest structure and the deepest link with other mathematical objects.
