A Note On Nilpotent Representations

TL;DR

This paper investigates the topology of representation and character varieties of finitely generated nilpotent groups into complex reductive groups, linking connected components to quotients by the lower central series.

Contribution

It provides a description of the topology of connected components of these varieties in terms of representations factoring through specific quotients.

Findings

Connected components characterized by lower central series quotients

Topology described in terms of factoring representations

Provides a framework for understanding representation varieties of nilpotent groups

Abstract

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.