#### Volume 19, issue 3 (2015)

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The topology of nilpotent representations in reductive groups and their maximal compact subgroups

### Maxime Bergeron

Geometry & Topology 19 (2015) 1383–1407
##### Abstract

Let $G$ be a complex reductive linear algebraic group and let $K\subset G$ be a maximal compact subgroup. Given a nilpotent group $\Gamma$ generated by $r$ elements, we consider the representation spaces $Hom\left(\Gamma ,G\right)$ and $Hom\left(\Gamma ,K\right)$ with the natural topology induced from an embedding into ${G}^{r}$ and ${K}^{r}$ respectively. The goal of this paper is to prove that there is a strong deformation retraction of $Hom\left(\Gamma ,G\right)$ onto $Hom\left(\Gamma ,K\right)$. We also obtain a strong deformation retraction of the geometric invariant theory quotient $Hom\left(\Gamma ,G\right)∕\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}∕G$ onto the ordinary quotient $Hom\left(\Gamma ,K\right)∕K$.

##### Keywords
strong deformation retraction, representation variety, character variety, nilpotent group, Kempf–Ness theory, geometric invariant theory, real and complex algebraic groups, maximal compact subgroup
##### Mathematical Subject Classification 2010
Primary: 20G20
Secondary: 55P99, 20G05
##### Publication
Received: 15 December 2013
Accepted: 25 July 2014
Published: 21 May 2015
Proposed: Benson Farb
Seconded: Walter Neumann, Martin R Bridson
##### Authors
 Maxime Bergeron Department of Mathematics The University of British Columbia Room 121 1984 Mathematics Road Vancouver, BC V6T 1Z2 Canada http://www.math.ubc.ca/~mbergeron/