A Note On Nilpotent Representations

Let $\Gamma$ be a finitely generated nilpotent group and let G be a complex reductive algebraic group. The representation variety $\mathrm{Hom}(\Gamma,G)$ and the character variety $\mathrm{Hom}(\Gamma,G)//G$ each carry a natural topology, and we describe the topology of their connected components in terms of representations factoring through quotients of $\Gamma$ by elements of its lower central series.