  In this thesis, we present the emergence of an $SL(2,R)$ ("Love") symmetry in
the suitably defined near-zone region, relevant for studying the black hole
response problem. This symmetry is globally defined and physical solutions of
the black hole linearized field equations are closed under its action. The
vanishing of static Love numbers is found to naturally arise as a selection
rule following from the fact that the relevant solution belongs to a particular
highest-weight representation of the Love symmetry. Interestingly, the Love
symmetry appears to be connected to the well-known enhanced $SL(2,R)$ isometry
subgroup of the near-horizon extremal geometry. Namely, the Love symmetry
exactly reduces to the isometry of the near-horizon AdS$_2$ throat for extremal
Reissner-Nordstr\"om black holes, while, for rotating black holes, one is lead
to consider an infinite-dimensional $SL(2,R)\ltimes\hat{U}(1)_V$ extension of
the Love symmetry, a family of subalgebras of which precisely recovers the
Killing vectors of the corresponding AdS$_2$ throat in the extremal limit.
Similar results persist when studying perturbations of general-relativistic
black holes in higher dimensions. Even though the black hole static Love
numbers now vanish only for a discrete set of resonant conditions that depends
on the orbital number of the perturbation, Love symmetry exists independently
of these details. This hints at a geometric interpretation of the Love
symmetry; a statement that becomes more rigorous within the framework of
subtracted geometries. However, Love symmetry appears to be theory-dependent.
We extract, in particular, a sufficient geometric condition for its existence
according to which, for instance, it does not exist in Riemann$^3$ gravity or
some stringy-corrected low-energy effective gravitational actions, in
accordance with explicit computations of the corresponding black hole Love
numbers.
