  We investigate asymptotic symmetries regularly defined on spherically
symmetric Killing horizons in the Einstein theory with or without the
cosmological constant. Those asymptotic symmetries are described by asymptotic
Killing vectors, along which the Lie derivatives of perturbed metrics vanish on
a Killing horizon. We derive the general form of asymptotic Killing vectors and
find that the group of the asymptotic symmetries consists of rigid O(3)
rotations of a horizon two-sphere and supertranslations along the null
direction on the horizon, which depend arbitrarily on the null coordinate as
well as the angular coordinates. By introducing the notion of asymptotic
Killing horizons, we also show that local properties of Killing horizons are
preserved under not only diffeomorphisms but also non-trivial transformations
generated by the asymptotic symmetry group. Although the asymptotic symmetry
group contains the $\mathit{Diff}(S^1)$ subgroup, which results from the
supertranslations dependent only on the null coordinate, it is shown that the
Poisson bracket algebra of the conserved charges conjugate to asymptotic
Killing vectors does not acquire non-trivial central charges. Finally, by
considering extended symmetries, we discuss that unnatural reduction of the
symmetry group is necessary in order to obtain the Virasoro algebra with
non-trivial central charges, which will not be justified when we respect the
spherical symmetry of Killing horizons.
