  For a stationary and axisymmetric black hole, there is a natural way to split
the fields into a probe sector and a background sector. The equations of motion
for the probe sector enjoy a significantly enhanced symmetry on the black hole
horizon. The extended symmetry is conformal in four dimensions, while in higher
dimensions it is much bigger. This puts conformal symmetry at the bottom of the
ladder of symmetries that can arise on black hole horizons in generic
dimensions.
