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

We describe the relation between G-character varieties, X_G(Γ), and G/H-character varieties, where H is a finite, central subgroup of G. In particular, we find finite generating sets for C[XG/H(Γ)] for classical groups G and H as above. By applying this approach to SO(4,C)=(SL(2,C)×SL(2,C))/Z/2 we find an explicit description of C[XSO(4,C)(F2)] for the free group on two generators, F₂.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[XG(Γ)] are generally not generated by the trace functions, τ_γ, for γ∈Γ, for G=SO(2n,C). In fact, we prove that the coordinate ring C[XSO(2n,C)(Γ)] is not even generated by "generalized trace functions," τ_γ,V, for all γ∈Γ and all representations V of SO(2n,C) for n=2 and groups Γ of corank ≥2.