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Poincare series of character varieties for nilpotent groups (1705.01443v1)
Published 3 May 2017 in math.AT, math.GT, and math.RT
Abstract: For any compact and connected Lie group $G$ and any free abelian or free nilpotent group $\Gamma$ , we determine the cohomology of the path component of the trivial representation of the representation space (character variety) $Rep(\Gamma,G)_1$, with coefficients in a field $F$ with ${char} (F)$ either 0 or relatively prime to the order of the Weyl group $W$. We give explicit formulas for the Poincar\'e 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 finitely generated free nilpotent group representations.