Exact solutions for extreme black hole magnetospheres

We present new exact solutions of Force-Free Electrodynamics (FFE) in the Near-Horizon region of an Extremal Kerr black hole (NHEK) and offer a complete classifica-tion of the subset that form highest-weight representations of the spacetime's SL(2, a"e)xU(1) isometry group. For a natural choice of spacetime embedding of this isometry group, the SL(2, a"e) highest-weight conditions lead to stationary solutions with non-trivial angular de-pendence, as well as axisymmetry when the U(1)-charge vanishes. In addition, we unveil a hidden SL(2, a",) symmetry of the equations of FFE that stems from the action of a complex automorphism group, and enables us to generate an SL(2, a",) family of (generically time-dependent) solutions. We then obtain still more general solutions with less symmetry by appealing to a principle of linear superposition that holds for solutions with collinear cur-rents. This allows us to resum the highest-weight primaries and their SL(2, a"e)-descendants.