Poincare series of character varieties for nilpotent groups

For any compact, connected Lie group G and any finitely generated nilpotent group Gamma, we determine the cohomology of the path component of the trivial representation of the group character variety (representation space) Rep(Gamma, G)(1), with coefficients in a field F with characteristic 0 or relatively prime to the order of the Weyl group W. We give explicit formulas for the Poincare series. In addition, we study G-equivariant stable decompositions of subspaces X(q, G) of the free monoid J(G) generated by the Lie group G, obtained from representations of finitely generated free nilpotent groups.