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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 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.

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
References
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/