Geometry & Topology

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

Maxime Bergeron

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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
Received: 15 December 2013
Accepted: 25 July 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
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

Subjects
Primary: 20G20: Linear algebraic groups over the reals, the complexes, the quaternions
Secondary: 55P99: None of the above, but in this section 20G05: Representation theory

Keywords
strong deformation retraction representation variety character variety nilpotent group Kempf–Ness theory geometric invariant theory real and complex algebraic groups maximal compact subgroup

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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