# Poincare series of character varieties for nilpotent groups

## TL;DR

This paper computes the cohomology and Poincaré series of certain character varieties associated with free nilpotent groups and compact Lie groups, providing explicit formulas and studying stable decompositions.

## Contribution

It introduces explicit formulas for the cohomology and Poincaré series of character varieties for free nilpotent groups, extending understanding of their topological structure.

## Key findings

- Explicit Poincaré series formulas derived
- Cohomology of character varieties determined
- Stable decompositions of related subspaces analyzed

## 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.