$sl(2,C)\times D$ symmetry and conformal primary basis for massless fields

Alternative to the embedding formalism, we provide a group theoretic approach to the conformal primary basis for the massless field with arbitrary helicity. To this end, we first point out that $sl(2,C)$ isometry gets enhanced to $sl(2,C)\times D$ symmetry for the solution space of the massless field with $D$ the bulk dilatation. Then, associated with $sl(2,C)\times D$ symmetry, we introduce the novel quadratic Casimirs and relevant tensor/spinor fields to derive two explicit constraints on the bulk dilatation and $sl(2,C)$ Casimirs. With this, we further argue that the candidate conformal primary basis can be constructed out of the infinite tower of the descendants of the left and right highest (lowest) conformal primary wave function of $sl(2,C)$ Lie algebra, and the corresponding celestial conformal weights are determined by the bulk scaling dimension through solving out the exact on-shell conformal primary wave functions, where on top of the two kinds of familiar-looking on-shell conformal primary wave functions, we also obtain another set of independent on-shell conformal primary wave functions for the massless field with helicity $\left|s\right|\ge 1$. In passing, we also develop the relationship between the four-dimensional Lorentz Lie algebra and two-dimensional conformal Lie algebra from scratch, and present an explicit derivation for the two important properties associated with the conformal primary wave functions.