Conformal Higher-Spin Symmetry for Matter Fields in $2+1$ Dimensions

A simple realization of the conformal higher-spin symmetry on the free massless matter fields in three dimensions is given in terms of an auxiliary Fock module both in the flat and in the $AdS_3$ case. The duality between nonunitary field theory representations of the conformal algebra and the unitary (singleton-type) representations of the $3d$ conformal algebra $sp(4,\mathcal R)$ is formulated explicitly in terms of a certain Bogoliubov transformation.