Convexes divisibles III

Let Δ0 be a group of finite type and FΔ0⊂Hom(Δ0,PGL(Rm)) be the subset of faithful representations for which there exists a properly convex Δ0-invariant open subset Ω in P(Rm) such that the quotient Δ0\Ω is compact. Koszul has proved in [J.L. Koszul, Déformation des connexions localement plates, Ann. Inst. Fourier 18 (1968) 103-114] that this subset FΔ0 is open. We describe the closure of FΔ0. As a consequence, we show that this subset FΔ0 is closed if and only if the virtual center of Δ0 is trivial. This condition is satisfied if and only if FΔ0 contains a strongly1 irreducible representation.