Representation varieties of the fundamental groups of non-orientable surfaces

Let $\Gamma_g$ be the fundamental group of a compact non-orientable surface of genus $g$ and let $K$ be an algebraically closed field of characteristic 0. The structure of the representation varieties $R(\Gamma_g,\mathrm{GL}_n(K))$, $R(\Gamma_g,\mathrm{SL}_n(K))$ of $\Gamma_g$ into $\mathrm{GL}_n(K)$ and $\mathrm{SL}_n(K)$ and of the character varieties $X(\Gamma_g,\mathrm{GL}_n(K))$ is described; namely, the number of their irreducible components and their dimensions are determined and their birational properties are investigated. It is proved, in particular, that all the irreducible components of $R(\Gamma_g,\mathrm{GL}_n(K))$ are $\mathbb Q$-rational varieties.