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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.

##### MSC:
 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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