It is well known that if a spacetime admits isometry, the space of eigenfunctions for the Laplacian is given as a representation of the isometry group provided an eigenvalue is fixed. In contrast, it has been less known how conformal symmetry of spacetime is related to the Laplacian. In this talk, we first revisit a relation between conformal symmetry of the two-dimensional sphere and ladder operators for the spherical harmonics. After that, we generalise the relation in a certain class of spacetimes which includes maximally symmetric spacetimes. As an application, we discuss the horizon instability of an extreme Reissner-Nordstrom black hole.