Using (the exponential form of) Fourier series, we can diagonalize the rotation operator $S_\theta$ that rotates complex-valued functions on $S^1$ by $\theta$ , $f(x) \mapsto f(x-\theta)$. Can we do the same thing on $S^2$? That is, can we diagonalize the operator $S_R$ that takes a function $f: S^2 \to \mathbb{C}$ and acts on it by a rotation R to get $f(R^{-1}\vec{x})$?
In fact, I'm interested in the case where $f$ is constant on lines of latitude, so that $f$ depends only on the polar angle. Can $S_R$ acting on such $f$ be diagonalized? (If it was already true in general, does the formula simplify for this case?)