Duality symmetries and G ⁺⁺⁺ theories

We show that the non-linear realisations of all the very extended algebras G⁺⁺⁺, except the B and C series which we do not consider, contain fields corresponding to all possible duality symmetries of the on-shell degrees of freedom of these theories. This result also holds for G_2⁺⁺⁺ and we argue that the non-linear realisation of this algebra accounts precisely for the form fields present in the corresponding supersymmetric theory. We also find a simple necessary condition for the roots to belong to a G⁺⁺⁺ algebra.