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

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, τγ, 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,” τγ,V, for all γ∈Γ and all representations V of SO(2n,C) for n=2 and groups Γ of corank ≥2.