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تسجيل مستخدم جديدPositivity and higher Teichmuller theory
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We introduce $Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $Theta$-positivity generalizes at the same time Lusztigs total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO(p,q) for p<q, and a family of exceptional Lie groups, which admit a $Theta$-positive structure. We describe key aspects of $Theta$-positivity and make a connection with representations of surface groups and higher Teichmuller theory.
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In arXiv:1802.02833 Guichard and Wienhard introduced the notion of $Theta$-positivity, a generalization of Lusztigs 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.
We introduce a new class of $mathfrak{sl}_2$-triples in a complex simple Lie algebra $mathfrak{g}$, which we call magical. Such an $mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $mathfrak{g}$. Using this decompo
sition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface $X$ for an associated real Lie group, hence also of the corresponding character variety of representations of $pi_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $mathrm{SO}_{p,q}$ with $1<pleq q$, and also constructs previously unknown components for the quaternionic real forms of $mathrm{E}_6$, $mathrm{E}_7$, $mathrm{E}_8$ and $mathrm{F}_4$. The classification of magical $mathfrak{sl}_2$-triples is shown to be in bijection with the set of $Theta$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmuller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Theta$-positive Anosov representations, which indicate that this conjecture holds.
Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by p
rimary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q> 3).
The goal of this article is to invite the reader to get to know and to get involved into higher Teichmuller theory by describing some of its many facets.
We prove a theorem that gives a sufficient condition for the full basic automorphism group of a complete Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category of Cartan foliations. Emphasize that the transverse C
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