Peter-Weyl tells us that the matrix coefficients of the irreps of $G$ are dense in $L^2(G)$. In the specific case of $G=SO(3)$, what are these coefficients?
Observation: Looking at the fundamental 3-dimensional irrep and the 5-dim irrep acting on traceless symmetric matrices, it seems that the matrix entries are related to but not the same as the spherical harmonics for the 3-dim, 5-dim irreps, respectively.
Question: Does this generalize? Are the matrix entries of the irreps of $SO(3)$ just spherical harmonics but somehow with more variables since $SO(3)$ has higher dimension than $S^2$?