Duality invariance in Fayet-Iliopoulos gauged supergravity

We propose a geometric method to study the residual symmetries in N = 2, d = 4 U(1) Fayet-Iliopoulos (FI) gauged supergravity. It essentially involves the stabilization of the symplectic vector of gauge couplings (FI parameters) under the action of the U-duality symmetry of the ungauged theory. In particular we are interested in those transformations that act non-trivially on the solutions and produce scalar hair and dyonic black holes from a given seed. We illustrate the procedure for finding this group in general and then show how it works in some specific models. For the prepotential F = −iX0X1, we use our method to add one more parameter to the rotating Chow-Compère solution, representing scalar hair.