Question. Why are the Wigner-D-matrices the appropriate representation for rotation of operators with respect to space-coordinates in physics?
Background. A physics paper brought up this question in my university group. Suppose for each $(x_1,\cdots,x_m) \in \mathbb{R}^{3m}$ (hence $x_i \in \mathbb{R}^3$ for every $i \in \{1,2,\cdots,m\}$), a matrix $H(x_1,\cdots,x_m)$ is studied that is equivariant with respect to $\text{SO}(3)$ in the following sense. If $q$ is a 3d rotation, then $$ H(qx_1,\cdots,qx_m) = \rho(q^*)H(x_1,\cdots,x_m) \rho(q). $$ The authors say that the representation $\rho$ is a Wigner matrix. In physics terminology, $H$ is a subblock of the hamiltonian and $(x_1,\cdots,x_m)$ are the coordinates of atom nuclei (but I don't expect specialist knowledge of atomic physics is necessary to understand the question). I have not found a 'Lie-theory'-explanation of why the Wigner-D-matrices appear in contexts such as these in physics. There is an audience for an answer of this type, because someone in my group said that in physics literature, the topic is often tackled by that the reader is given a parameterization of the Wigner matrices and told to calculate, so to speak. I feel the same way, and therefore an explanation from a graduate-level (or higher) in mathematics would be helpful. Why is it the appropriate representation, and not something like the 'defining representation of SO(3)', for example (the one given by the inclusion)?