Positivity and representations of surface groups

Abstract

In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Θ$-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 $Θ$-positive representations of surface groups. We prove that $Θ$-positive representations are $Θ$-Anosov. This implies that $Θ$-positive representations are discrete and faithful and that the set of $Θ$-positive representations is open in the representation variety. We show that the set of $Θ$-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 $Θ$-positive structure there exist components consisting of $Θ$-positive representations. More precisely we prove that the components parametrized using Higgs bundles methods in arXiv:2101.09377 consist of $Θ$-positive representations.