Astronomical observation suggests the existence of near-extreme Kerr black holes whose horizons spin at nearly the speed of light. Properties of diffeomorphisms imply that the dynamics of the high-redshift near-horizon region of near-extreme Kerr, which includes the innermost-stable-circular-orbit (ISCO), is governed by an infinite-dimensional emergent conformal symmetry. This symmetry may be exploited to analytically, rather than numerically, compute a variety of potentially observable processes. In this talk I will show how we compute and study the conformal transformation properties of the gravitational radiation emitted by an orbiting massive object in the large-redshift near-horizon region. I will also use conformal symmetry of the near-horizon region to compute the gravitational radiation produced during the plunge phase following the object's crossing of the ISCO.