an:01488567
Zbl 0957.22008
Benoist, Yves
Automorphisms of convex cones
FR
Invent. Math. 141, No. 1, 149-193 (2000).
00066339
2000
j
22E15 22E46 20G20
properly convex cone; Zariski closure; semisimple Lie group; irreducible representation
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.
Vasile Oproiu (Ia??i)