Infinitesimal Zariski closures of positive representations (Q6633911)

This paper classifies Zariski-closures of positive representations. For a simple split real Lie group \(G\), its full flag variety \(\mathcal{F}\) carries a notion of positive configuration of tuples of flags. Given a closed hyperbolic surface \(S\), a representation \(\rho:\pi_1(S)\to G\) is then said to be positive if there is a continuous \(\rho\)-equivariant map \(\xi:\partial_{\infty}\pi_1(S)\to\mathcal{F}\) that maps triple of distinct points in the boundary of the group to positive triples of flags. As shown by \textit{V. Fock} and \textit{A. Goncharov} [Publ. Math., Inst. Hautes Étud. Sci. 103, 1--211 (2006; Zbl 1099.14025)], positive representations coincide with Hitchin representations which make them a central object of study in higher Teichmüller theory.\N\NMore generally, this paper classifies the semisimple part of the Zariski-closure of partially positive representations. Let \(X\) be a proper Gromov-hyperbolic space and \(\Gamma<\mathrm{Isom}(X)\) a discrete subgroup with limit set \(\partial X_{\Gamma}\) in the visual boundary of \(X\). A representation \(\rho:\Gamma\to G\) is partially positive if there exists a continuous \(\rho\)-equivariant map \(\xi:\partial X_{\Gamma}\to \mathcal{F}\) such that for every pair of distinct points \(x,z\in\partial X_{\Gamma}\), there exists a third point \(y\in\partial X_{\Gamma}\) such that \((\xi(x),\xi(y),\xi(z))\) is a positive triple. Cusped Hitchin representations of non-elementary geometrically finite Fuchsian groups are examples of partially positive representations.\N\NThe main tool of the proof is the Hasse diagram associated to a representation of a semisimple Lie group. The vertices of this diagram are the weights of the representation and two weights are joined by an edge if they differ by a simple root. Let \(H<G\) be the semisimple part of the Zariski-closure of a partially positive representation. The author constructs a surjective map \(f:\Delta_G\to\Delta_H\) from the Dynkin diagram of \(G\) to the Dynkin diagram of \(H\). For any \(\alpha\in\Delta_G\), \(f\) induces a surjective diagram map from the Hasse diagram of the \(\alpha\)-fundamental representation of \(G\) to the Hasse diagram of the \(f(\alpha)\)-fundamental representation of \(H\). A case-by-case study of the relevant Hasse diagrams yields the classification of such \(H\).