  We prove that for every reductive group G with a maximal torus T and the Weyl
group W there is a natural normalization map chi from T^N/W to an irreducible
component of the G-character variety of Z^N. We prove that chi is an
isomorphism for all classical groups.
  Additionally, we prove that even though there are no irreducible
representations in the above mentioned irreducible component of the character
variety for non-abelian G, the tangent spaces to it coincide with H^1(Z^N, Ad
rho). Consequently, this irreducible component has the "Goldman" symplectic
form for N=2, for which the combinatorial formulas for Goldman bracket hold.
