## Geometry & Topology

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

Maxime Bergeron

#### Abstract

Let $G$ be a complex reductive linear algebraic group and let $K ⊂ G$ be a maximal compact subgroup. Given a nilpotent group $Γ$ generated by $r$ elements, we consider the representation spaces $Hom(Γ,G)$ and $Hom(Γ,K)$ with the natural topology induced from an embedding into $Gr$ and $Kr$ respectively. The goal of this paper is to prove that there is a strong deformation retraction of $Hom(Γ,G)$ onto $Hom(Γ,K)$. We also obtain a strong deformation retraction of the geometric invariant theory quotient $Hom(Γ,G)∕∕G$ onto the ordinary quotient $Hom(Γ,K)∕K$.

#### Article information

Source
Geom. Topol., Volume 19, Number 3 (2015), 1383-1407.

Dates
Accepted: 25 July 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858765

Digital Object Identifier
doi:10.2140/gt.2015.19.1383

Mathematical Reviews number (MathSciNet)
MR3352239

Zentralblatt MATH identifier
1346.20063

#### Citation

Bergeron, Maxime. The topology of nilpotent representations in reductive groups and their maximal compact subgroups. Geom. Topol. 19 (2015), no. 3, 1383--1407. doi:10.2140/gt.2015.19.1383. https://projecteuclid.org/euclid.gt/1510858765

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