A Note On Nilpotent Representations

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