Symmetries at the black hole horizon

We determine the asymptotic symmetry group of Killing horizons by choosing Gaussian null coordinates in the neighborhood of the horizon and boundary conditions that respect the leading-order terms in the metric. The analysis divides naturally into the two cases of subextremal and extremal horizons. In general, we find rather involved asymptotic symmetry generators that nevertheless involve supertranslations and Killing vectors on the compact horizon. The most striking observation is the difference in the dependence on the coordinate along the horizon; we relate this to the redshift effect for subextremal horizons. We consider the spherically symmetric case as a special and illuminating example.