G-Character varieties for G=SO(n,C) and other not simply connected groups

We describe the relation between G-character varieties, $X_G(\Gamma)$, and $G/H$-character varieties, where $H$ is a finite, central subgroup of $G.$ In particular, we find finite generating sets of coordinate rings $C[X_{G/H}(\Gamma)]$ for classical groups $G$ and $H$ as above. Using this approach we find an explicit description of $C[X_{SO(4,C)}(F_2)]$ for the free group on two generators, $F_2.$ In the second part of the paper, we prove several properties of SO(2n,C)-character varieties. This is a particularly interesting class of character varieties because unlike for all other classical groups G, the coordinate rings $C[X_{G}(\Gamma)]$ are generally not generated by trace functions $\tau_\gamma$, for $\gamma\in \Gamma$, for G=SO(2n,C). In fact, we prove that the coordinate ring $C[X_{SO(2n,C)}(\Gamma)]$ is not even generated by "generalized trace functions," $\tau_{\gamma,V},$ for all $\gamma\in \Gamma$ and all representations $V$ of $SO(2n,C)$ for $n=2$ and groups $\Gamma$ of corank $\geq 2$.