Character Varieties of Abelian Groups

We prove that for every reductive group G with a maximal torus T and Weyl group W there is a natural normalization map χ from T/W to an irreducible component, X 0 G(Z ), of the G-character variety XG(Z ) of Z . We prove that this component is reduced and that χ is an isomorphism for classical groups. Additionally, we prove that even though there are no irreducible representations of Z into reductive groups of rank > 1, the tangent spaces to X 0 G(T ), at generic points for T torus, have cohomological interpretation analogous to that for closed surfaces of higher genus. Consequently, X 0 G(T ), has the “Goldman” symplectic form for which combinatorial formulas for Goldman bracket hold.