  We compute the radiation emitted by a particle on the innermost stable
circular orbit of a rapidly spinning black hole both (a) analytically, working
to leading order in the deviation from extremality and (b) numerically, with a
new high-precision Teukolsky code. We find excellent agreement between the two
methods. We confirm previous estimates of the overall scaling of the power
radiated, but show that there are also small oscillations all the way to
extremality. Furthermore, we reveal an intricate mode-by-mode structure in the
flux to infinity, with only certain modes having the dominant scaling. The
scaling of each mode is controlled by its conformal weight, a quantity that
arises naturally in the representation theory of the enhanced near-horizon
symmetry group. We find relationships to previous work on particles orbiting in
precisely extreme Kerr, including detailed agreement of quantities computed
here with conformal field theory calculations performed in the context of the
Kerr/CFT correspondence.
