Geometry & Topology

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

Maxime Bergeron

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

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  • A Ádem, F R Cohen, Commuting elements and spaces of homomorphisms, Math. Ann. 338 (2007) 587–626
  • M Bergeron, L Silberman, A note on nilpotent representations
  • A Borel, Linear algebraic groups, 2nd edition, Graduate Texts in Mathematics 126, Springer, New York (1991)
  • A Borel, R Friedman, J W Morgan, Almost commuting elements in compact Lie groups, Mem. Amer. Math. Soc. 747, Amer. Math. Soc. (2002)
  • M Brion, Introduction to actions of algebraic groups, Les cours du CIRM 1, CIRM (2010) 1–22
  • J Dixmier, W G Lister, Derivations of nilpotent Lie algebras, Proc. Amer. Math. Soc. 8 (1957) 155–158
  • J L Dyer, A nilpotent Lie algebra with nilpotent automorphism group, Bull. Amer. Math. Soc. 76 (1970) 52–56
  • C Florentino, S Lawton, The topology of moduli spaces of free group representations, Math. Ann. 345 (2009) 453–489
  • C Florentino, S Lawton, Topology of character varieties of abelian groups, Topology Appl. 173 (2014) 32–58
  • J M Gómez, A Pettet, J Souto, On the fundamental group of ${\rm Hom}({\mathbb Z}\sp k,G)$, Math. Z. 271 (2012) 33–44
  • R Grone, C R Johnson, E M Sa, H Wolkowicz, Normal matrices, Linear Algebra Appl. 87 (1987) 213–225
  • R A Horn, C R Johnson, Matrix analysis, revised 2nd edition, Cambridge Univ. Press (1990)
  • L C Jeffrey, Flat connections on oriented $2$–manifolds, Bull. London Math. Soc. 37 (2005) 1–14
  • V G Kac, A V Smilga, Vacuum structure in supersymmetric Yang–Mills theories with any gauge group, from: “The many faces of the superworld”, (M Shifman, editor), World Sci. Publ., Singapore (2000) 185–234
  • G Kempf, L Ness, The length of vectors in representation spaces, from: “Algebraic geometry”, (K Lønsted, editor), Lecture Notes in Math. 732, Springer, Berlin (1979) 233–243
  • S Lojasiewicz, Ensembles semi-analytiques, preprint, Institut des Haute Études Scientifiques (1965)
  • A Lubotzky, A R Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 336, Amer. Math. Soc. (1985)
  • D Luna, Sur certaines opérations différentiables des groupes de Lie, Amer. J. Math. 97 (1975) 172–181
  • D Luna, Fonctions différentiables invariantes sous l'opération d'un groupe réductif, Ann. Inst. Fourier $($Grenoble$)$ 26 (1976) 33–49
  • G D Mostow, Self-adjoint groups, Ann. of Math. 62 (1955) 44–55
  • M S Narasimhan, C S Seshadri, Holomorphic vector bundles on a compact Riemann surface, Math. Ann. 155 (1964) 69–80
  • A Neeman, The topology of quotient varieties, Ann. of Math. 122 (1985) 419–459
  • A L Onishchik, È B Vinberg, Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer, Berlin (1990)
  • A Pettet, J Souto, Commuting tuples in reductive groups and their maximal compact subgroups, Geom. Topol. 17 (2013) 2513–2593
  • M S Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer, New York (1972)
  • R W Richardson, Conjugacy classes of $n$–tuples in Lie algebras and algebraic groups, Duke Math. J. 57 (1988) 1–35
  • R W Richardson, P J Slodowy, Minimum vectors for real reductive algebraic groups, J. London Math. Soc. 42 (1990) 409–429
  • G W Schwarz, The topology of algebraic quotients, from: “Topological methods in algebraic transformation groups”, (H Kraft, T Petrie, G W Schwarz, editors), Progr. Math. 80, Birkhäuser, Boston (1989) 135–151
  • A S Sikora, Character varieties, Trans. Amer. Math. Soc. 364 (2012) 5173–5208
  • C T Simpson, Moduli of representations of the fundamental group of a smooth projective variety, I, Inst. Hautes Études Sci. Publ. Math. (1994) 47–129
  • E Witten, Constraints on supersymmetry breaking, Nuclear Phys. B 202 (1982) 253–316
  • E Witten, Toroidal compactification without vector structure, J. High Energy Phys. (1998) Paper 6, 43 pp.