lie algebra - Derivation of the irreducible representations of SO(3)

Is there a way to derive the representations of $SO(3)$ without the usual method with the ladder operators which also gives the ones of $SU(2)$?

The usual way to do these calculations is to start from the commutation relations of the Lie algebra associated with $SU(2)$ (or that of $SO(3)$, which is the same given that $\mathfrak{so}(3) \approx \mathfrak{su}(2)$ as far as I understand) and from there to go throught the ladder-operators-thing to obtain all of the representations of $SU(2)$. Is there another way to derive the representations of $SO(3)$ which is specific of $SO(3)$ and not also applicable to $SU(2)$?

Answer

Be careful. It may be the case that $\mathfrak{su}(2)=\mathfrak{so}(3)$, but it is not the case that $SU(2)=SO(3)$. $SU(2)=\mathrm{Spin}(3)$ and $\rho :SU(2)\rightarrow SO(3)$ is the two-sheeted universal cover of $SO(3)$. It thus turns out that only the integer spin representations of $SU(2)$ factor through $\rho$ to give well-defined representations of $SO(3)$.

Concretely, the spherical harmonics $Y_{\ell m}(\theta ,\phi )$ span an irreducible representation of $SO(3)$ in $L^2(S^2)$. This representation has spin $\ell$ (dimension $2\ell +1$), and this yields all the finite-dimensional irreducible complex representations of $SO(3)$. Note that $\ell$ must be an integer in this context (and so it does not give you all the irreducible representations of $SU(2)$).