  In this note, we present a synopsis of geometric symmetries for (spin 0)
perturbations around (4D) black holes and de Sitter space. For black holes, we
focus on static perturbations, for which the (exact) geometric symmetries have
the group structure of SO(1,3). The generators consist of three spatial
rotations, and three conformal Killing vectors obeying a special melodic
condition. The static perturbation solutions form a unitary (principal series)
representation of the group. The recently uncovered ladder symmetries follow
from this representation structure; they explain the well-known vanishing of
the black hole Love numbers. For dynamical perturbations around de Sitter
space, the geometric symmetries are less surprising, following from the SO(1,4)
isometry. As is well known, the quasinormal solutions form a non-unitary
representation of the isometry group. We provide explicit expressions for the
ladder operators associated with this representation. In both cases, the ladder
structures help connect the boundary condition at the horizon with that at
infinity (black hole) or origin (de Sitter space), and they manifest as
contiguous relations of the hypergeometric solutions.
