Character varieties of abelian groups

We prove that for every reductive group G with a maximal torus T and the Weyl group W, T^N/W is the normalization of the irreducible component, X⁰_G(ℤ^N), of the G-character variety X_G(ℤ^N) of ℤ^N containing the trivial representation. We also prove that X⁰_G(ℤ^N)=T^T/W for all classical groups. Additionally, we prove that even though there are no irreducible representations in X⁰_G(ℤ^N) for non-abelian G, the tangent spaces to X⁰_G(ℤ^N) coincide with H¹(ℤ^N, Ad ρ). Consequently, X⁰_G(ℤ²), has the "Goldman" symplectic form for which the combinatorial formulas for Goldman bracket hold.