  Perturbations of massless fields in the Kerr-Newman black hole background
enjoy a (``Love'') SL$(2,\mathbb{R})$ symmetry in the suitably defined near
zone approximation. We present a detailed study of this symmetry and show how
the intricate behavior of black hole responses in four and higher dimensions
can be understood from the SL$(2,\mathbb{R})$ representation theory. In
particular, static perturbations of four-dimensional black holes belong to
highest weight SL$\left(2,\mathbb{R}\right)$ representations. It is this
highest weight property that forces the static Love numbers to vanish. We find
that the Love symmetry is tightly connected to the enhanced isometries of
extremal black holes. This relation is simplest for extremal charged
spherically symmetric (Reissner-Nordstr\"om) solutions, where the Love symmetry
exactly reduces to the isometry of the near horizon AdS$_2$ throat. For
rotating (Kerr-Newman) black holes one is lead to consider an
infinite-dimensional SL$\left(2,\mathbb{R}\right)\ltimes \hat
U(1)_{\mathcal{V}}$ extension of the Love symmetry. It contains three
physically distinct subalgebras: the Love algebra, the Starobinsky near zone
algebra, and the near horizon algebra that becomes the Bardeen-Horowitz
isometry in the extremal limit. We also discuss other aspects of the Love
symmetry, such as the geometric meaning of its generators for spin weighted
fields, connection to the no-hair theorems, non-renormalization of Love
numbers, its relation to (non-extremal) Kerr/CFT correspondence and prospects
of its existence in modified theories of gravity.
