TITLE: Spherical harmonics give all the irreducible representations of $SO(3)$?
QUESTION [8 upvotes]: It is mentioned in Wiki that the spaces $\mathcal{H}_{k}$ of spherical harmonics of degree $k$ give ALL the irreducible representations of $SO(3)$. Could anyone tell me where can I find the proof? Thanks!
EDIT: I am seeking for an elementary proof that dose not require too many big machineries in representation theory.

REPLY [2 votes]: The proof is rather simple, just calculate characters (for rotations around OZ, since the axis does not affect the character) , and, using orthogonality theorem, note that all Fourier series coefficient for any other character are zero. But functions $\cos (l*\phi) (l - n)$, (where $n$ is an integer) form a complete set on $<0, \pi>$, so there are no more irreducible, unequivocal representations of $SO(3)$.