  $J\bar T$-deformed CFTs provide an interesting example of non-local, yet
UV-complete two-dimensional QFTs that are entirely solvable. They have been
recently shown to possess an infinite set of symmetries, which are a continuous
deformation of the Virasoro-Kac-Moody symmetries of the seed CFT. In this
article, we put forth a definition of primary operators in $J\bar T$-deformed
CFTs on a cylinder, which are singled out by having CFT-like momentum-space
commutation relations with the symmetry generators in the decompatification
limit. We show -- based on results we first derive for the case of $J^1 \wedge
J^2$-deformed CFTs -- that all correlation functions of such operators in the
$J\bar T$-deformed CFT can be computed exactly in terms of the correlation
functions of the undeformed CFT and are crossing symmetric in the plane limit.
In particular, two and three-point functions are simply given by the
corresponding momentum-space correlator in the undeformed CFT, with all
dimensions replaced by particular momentum-dependent conformal dimensions.
Interestingly, scattering amplitudes off the near-horizon of extremal black
holes are known to take a strikingly similar form.
