We study asymptotically flat stationary solutions of four-dimensional supergravity theories via the associated G/H * pseudo-Riemannian non-linear sigma models in three spatial dimensions. The Noether charge C associated to G is shown to satisfy a characteristic equation that determines it as a function of the four-dimensional conserved charges. The matrix C is nilpotent for non-rotating extremal solutions. The nilpotency degree of C is directly related to the BPS degree of the corresponding solution when they are BPS.Equivalently, the charges can be described in terms of a Weyl spinor |C of Spin * (2N ), and then the characteristic equation becomes equivalent to a generalisation of the Cartan pure spinor constraint on |C . The invariance of a given solution with respect to supersymmetry is determined by an algebraic 'Dirac equation' on the Weyl spinor |C . We explicitly solve this equation for all pure supergravity theories and we characterise the stratified structure of the moduli space of asymptotically Taub-NUT black holes with respect to their BPS degree. The analysis is valid for any asymptotically flat stationary solutions for which the singularities are protected by horizons. The H * -orbits of extremal solutions are identified as Lagrangian submanifolds of nilpotent orbits of G, and so the moduli space of extremal spherically symmetric black holes is identified as a Lagrangian subvariety of the variety of nilpotent elements of g. We also generalise the notion of active duality transformations to an 'almost action' of the three-dimensional duality group G on asymptotically flat stationary solutions. * email address: bossard@aei.mpg.de † email address: Hermann.Nicolai@aei.mpg.de ‡ email address: k.stelle@imperial.ac.uk where the 'momentum' ǫ is the asymptotic supersymmetry parameter (Killing spinor)

We investigate and clarify the mutual compatibility of the higher order corrections arising in supergravity and string theory effective actions and the non-linear duality symmetries of these theories. Starting from a conventional tree level action leading to duality invariant equations of motion, we show how to accommodate duality invariant counterterms given as functionals of both electric and magnetic fields in a perturbative expansion, and to deduce from them a non-polynomial bona fide action satisfying the Gaillard-Zumino (NGZ) constraint. There exists a corresponding consistency constraint in the non-covariant Henneaux-Teitelboim formalism which ensures that one can always restore diffeomorphism invariance by perturbatively solving this functional identity. We illustrate how this procedure works for the R 2 ∇F ∇F and F 4 counterterms in Maxwell theory.