Positivity and representations of surface groups
In arXiv:1802.02833 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 prov...
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| Main Authors: | , , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
16 Aug 2021
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| In: |
Arxiv
Year: 2021, Pages: 1-42 |
| DOI: | 10.48550/arXiv.2106.14584 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2106.14584 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2106.14584 
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| Author Notes: | Olivier Guichard, François Labourie, and Anna Wienhard |
MARC
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| 520 | |a In arXiv:1802.02833 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:2101.09377 consist of $\Theta$-positive representations. | ||
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