I am aware that in certain cases, the irreducible representations of a group are equivalent to the harmonics of the homogeneous space for that group. For instance, the irreducible representations of $\mathbb{S}1$ are equivalent to the harmonics of the ring. Likewise, the irreducible representations of $SO(3)$ are equivalent to the harmonics of the sphere.
Could someone explain the more general relationship between these objects? And why there is this equivalence in the examples of $\mathbb{S}1$ and $SO(3)$?