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Pré-Publication, Document De Travail Année : 2021

Positivity and representations of surface groups

Olivier Guichard
  • Fonction : Auteur
Anna Wienhard
  • Fonction : Auteur

Résumé

In arXiv:1802.0283(3) Guichard and Wienhard introduced the notion of $\Theta$-positivity, a generalization of Lusztig's total positivity to real Lie groups that are not necessarily split. Based on this notion, we introduce in this paper $\Theta$-positive representations of surface groups. We prove that $\Theta$-positive representations are $\Theta$-Anosov. This implies that $\Theta$-positive representations are discrete and faithful and that the set of $\Theta$-positive representations is open in the representation variety. We show that the set of $\Theta$-positive representations is closed within the set of representations that do not virtually factor through a parabolic subgroup. From this we deduce that for any simple Lie group $\mathsf G$ admitting a $\Theta$-positive structure there exist components consisting of $\Theta$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.0937(7) consist of $\Theta$-positive representations.

Dates et versions

hal-03273224 , version 1 (29-06-2021)

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Olivier Guichard, François Labourie, Anna Wienhard. Positivity and representations of surface groups. 2021. ⟨hal-03273224⟩
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