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We show that any static linear perturbations around Schwarzschild black holesenjoy a set of conserved charges which forms a centrally extendedSchr\"{o}dinger algebra sh(1) = sl$(2,\mathbb{R}) \ltimes \mathcal{H}$. Thecentral charge is given by the black hole mass, echoing results on black holeentropy from near-horizon diffeomorphism symmetry. The finite symmetrytransformations generated by these conserved charges correspond to Galilean andconformal transformations of the static field and of the coordinates. This newstructure allows one to discuss the static response of a Schwarzschild blackhole in the test field approximation from a symmetry-based approach. First weshow that the (horizontal) symmetry protecting the vanishing of the Lovenumbers recently exhibited by Hui et al, dubbed the HJPSS symmetry, coincideswith one of the sl$(2,\mathbb{R})$ generators of the Schr\"{o}dinger group.Then, it is demonstrated that the HJPSS symmetry is selected thanks to thespontaneous breaking of the full Schr\"{o}dinger symmetry at the horizon downto a simple abelian sub-group. The latter can be understood as the symmetryprotecting the regularity of the test field at the horizon. In the4-dimensional case, this provides a symmetry protection for the vanishing ofthe Schwarzschild Love numbers. Our results trivially extend to the Kerr case.

Hidden symmetry of the static response of black holes: Applications to Love numbers