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Automorphisms of convex cones. (Automorphismes des cônes convexes.) (French) Zbl 0957.22008
The author studies the subgroups of \(GL(m,\mathbb{R})\) preserving a properly convex cone of \(\mathbb{R}^m\) and whose action on \(\mathbb{R}^m\) is irreducible. Let \(\Gamma\) be a subgroup of \(GL(m,\mathbb{R})\), preserving the properly convex cone \(C\subset\mathbb{R}^m\). If \(C\) is strictly convex and \(\Gamma \setminus C\) is compact then the Zariski closure \(G\) of \(\Gamma\) is either \(GL(m,\mathbb{R})\) or the similitude subgroup of a Lorentzian quadratic form on \(\mathbb{R}^m\). Then one describes the Zariski closure \(G\) of \(\Gamma\) under the hypothesis that the action on \(\mathbb{R}^m\) is irreducible. It follows that \(G\) is a semisimple Lie group and \(\mathbb{R}^m\) is an irreducible representation of \(G\). The irreducible representations of this kind are characterized by the following properties: the representation is proximal and the highest weight \(\lambda\) does not coincide “modulo 2” with the restricted highest weight of an irreducible symplectic proximal representation. The results are used to describe the group \(G\) corresponding to a group \(\Gamma\) for which all the eigenvalues are strictly positive, e.g. \(G= GL(m, \mathbb{R})\) if and only if \(m\neq 2\), modulo 4.

22E15 General properties and structure of real Lie groups
22E46 Semisimple Lie groups and their representations
20G20 Linear algebraic groups over the reals, the complexes, the quaternions
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