This might be more of a soft question, since I don't While learning about representations of the Lorentz group, I found in Maggiore's book (Chapter 2) that massive particles of spin $j$ have $2j+1$ degrees of freedom, whereas massless particles only have one degree of freedom, the helicity. That's why a photon, which we often call a massless spin-1 particle, has two polarizations and its "spin" (helicity really) can only take two values.
The argument follows from having two Casmir operators for the Lorentz group, $P^\mu P_\mu, W^\mu W_\mu$, with $W^\mu$ being the Pauli-Lubanski 4-vector, which commute. After that, we make the distiction between $P^\mu P_\mu=m=0$ and $m\neq 0$. So, then $m=0$ reduces the degrees of freedom of the field that transforms under the Lorentz group. Please, do correct me if I'm wrong.
This might be a bit generic but, in the case of an internal symmetry (like a gauge symmetry), can there be an analogous restriction to the degrees of freedom? If so, are there any examples of models that do that?
As an example, I'm thinking about a theory with internal symmetry under a group $G$, with its Casimirs, $C_1, C_2, .. C_n$, with one $C_j$ (analogous to $P^\mu P_\mu$ for the Lorentz group) such that $C_j=0$ (with $C_j$ not being trivial).