On invariant structures of black hole charges

We study minimal degree complete bases of duality- and horizontal - invariant homogeneous polynomials in the flux representation of two-centered black hole solutions in two classes of D = 4 Einstein supergravity models with symmetric vector multiplets' scalar manifolds. Both classes exhibit an SL (2, R) horizontal symmetry which mixes the two centers. The first class encompasses N = 2 and N = 4 matter-coupled theories, with semi-simple U-duality given by SL (2, R) x SO (m, n); the analysis is carried out in the so-called Calabi-Vesentini symplectic frame (exhibiting maximal manifest covariance) and until order six in the fluxes included. The second class, exhibiting a non-trivial horizontal stabilizer SO(2), includes N = 2 minimally coupled and N = 3 matter coupled theories, with U-duality given by the pseudounitary group U(r, s) (related to complex flux representations). Finally, we comment on the formulation of special Kahler geometry in terms of generalized groups of type E-7.