I'm reading the following paragraph of these lecture notes (pp 35):
I'm trying to understand the difference in defining spherical harmonics as functions on $S^2$ or $SO(3)$. From what I've read we can see $S^2$ as homogeneous space of $SO(3)$, this means that we can identify $S^2 \cong SO(3)/H$ where $H$ is a subgroup of $SO(3)$ that maintains fixed a chosen point under rotations (i.e. $H = Stab_{x_0}=\{g \in SO(3) \, | \, gx_0 = x_0\}$). But actually here we are considering the full $SO(3)$ group as domain for our spherical harmonics.