Bad Representations and Homotopy of Character Varieties
Annales Henri Lebesgue, Volume 5 (2022), pp. 93-140.

Let G be a connected reductive complex affine algebraic group, and let 𝔛 r denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in 𝔛 r , extending a theorem of Richardson and proving a Mumford-type result about topological singularities; this resolves conjectures of Florentino–Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel–de Siebenthal subgroups. We then use the codimension bound to calculate higher homotopy groups of the smooth locus of 𝔛 r , proving conjectures of Florentino–Lawton–Ramras. Lastly, using the earlier analysis of Borel–de Siebenthal subgroups, we prove a conjecture of Sikora about centralizers of irreducible representations in Lie groups.

Soit G un groupe algébrique complexe réductif connexe et 𝔛 r l’espace des modules de représentations d’un groupe libre de rang r dans G. Nous commençons par caractériser les singularités de 𝔛 r et nous montrons en particulier que les voisinages de ces singularités sont des singularités topologiques. Ceci élucide une conjecture émise par Florentino–Lawton. L’étude des sous-groupes de Borel–de Siebenthal nous permet alors de calculer la codimension du lieu singulier orbifold de 𝔛 r . On utilise ensuite ce calcul pour déterminer les groupes d’homotopies du lieu lisse de 𝔛 r . Ceci démontre des conjectures de Florentino–Lawton–Ramras. Finalement, en appliquant la classification des sous-groupes de Borel–de Siebenthal, nous démontrons une conjecture de Sikora concernant les centralisateurs de représentations irréductibles dans les groupes de Lie.

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DOI: 10.5802/ahl.119
Classification: 14B05, 14L24, 55Q05, 14D20, 14L30, 55U10
Keywords: character variety, Borel-de Siebenthal subgroups, free group, homotopy groups, singularities
Guérin, Clément 1; Lawton, Sean 2; Ramras, Daniel 3

1 Science and Technology Department, Mayotte University Center, Route nationale 3, BP 53, 97660 Dembeni, (France)
2 Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, Virginia 22030, (USA)
3 Department of Mathematical Sciences, Indiana University-Purdue University Indianapolis, 402 N. Blackford, Indianapolis, IN 46202, (USA)
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Guérin, Clément; Lawton, Sean; Ramras, Daniel. Bad Representations and Homotopy of Character Varieties. Annales Henri Lebesgue, Volume 5 (2022), pp. 93-140. doi : 10.5802/ahl.119. http://www.numdam.org/articles/10.5802/ahl.119/

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