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.