Positivity and representations of surface groups

Abstract

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

There was a wrong statement in the appendix of the previous versions. The statement of the main result has therefore to be altered