  Kinematic space has been defined as the space of codimension-$2$ spacelike
extremal surfaces in anti de Sitter (AdS$_{d+1}$) spacetime which, by the
Ryu-Takayanagi proposal, compute the entanglement entropy of spheres in the
boundary CFT$_d$. It has recently found many applications in holography.
Coadjoint orbits are symplectic manifolds that are the classical analogues of a
Lie group's unitary irreducible representations. We prove that kinematic space
is a particular coadjoint orbit of the $d$-dimensional conformal group
$SO(d,2)$. In addition, we show that the Crofton form on kinematic space
associated to AdS$_3$, that was shown to compute the lengths of bulk curves, is
equal to the standard Kirillov-Kostant symplectic form on the coadjoint orbit.
Since kinematic space is K\"{a}hler in addition to symplectic, it can be
quantized. The orbit method extends the kinematic space dictionary, which was
originally motivated through connections to integral geometry, by directly
translating geometrical properties of holographic auxiliary spaces into
statements about the representation theory of the conformal group.
