  We consider quantum field theory near the horizon of an extreme Kerr black
hole. In this limit, the dynamics is well approximated by a tower of
electrically charged fields propagating in an $SL(2,\mathbb{R})$ invariant
AdS$_2$ geometry endowed with a constant, symmetry preserving background
electric field. At large charge the fields oscillate near the AdS$_2$ boundary
and no longer admit a standard Dirichlet treatment. From the Kerr black hole
perspective, this phenomenon is related to the presence of an ergosphere. We
discuss a definition for the quantum field theory whereby we 'UV' complete
AdS$_2$ by appending an asymptotically two dimensional Minkowski region. This
allows the construction of a novel observable for the flux-carrying modes that
resembles the standard flat space S-matrix. We relate various features
displayed by the highly charged particles to the principal series
representations of $SL(2,\mathbb{R})$. These representations are unitary and
also appear for massive quantum fields in dS$_2$. Both fermionic and bosonic
fields are studied. We find that the free charged massless fermion is exactly
solvable for general background, providing an interesting arena for the problem
at hand.
