I‘m having trouble understanding what exactly the irreducible representations of $\operatorname{SO}(3)$ are.
I know that the irreducible representations of $\operatorname{SU}(2)$ are given by:
$\rho(A)p(x)=p(A^{-1}x)$, where $A \in\operatorname{SU}(2)$ and $p(x) \in V_l$ with $V_l$ the vector space of homogenous polynomials in 2 variables of degree $l$ (dim$V_l=l+1$). So there‘s one irred for every dimension.
No there is a surjective homomorphism $\phi:\operatorname{SU}(2) \rightarrow\operatorname{SO}(3)$ with $\ker \phi=\{ \pm \mathbb{1}\}$, this means that for even $l$ and for $B \in\operatorname{SO}(3)$ we find and irreducible representation of $\operatorname{SO}(3)$: $ B \mapsto\rho(\phi^{-1}B)$, which is well defined because $(-x_1)^n(-x_2)^m= (x_1)^n(x_2)^m $ for $n+m$ even.
Now my question is whether these are all irreds of $\operatorname{SO}(3)$, because I have also seen that, we can construct irreds of $\operatorname{SO}(3)$ on the space of harmonic, homogenous polynomials of degree $l$ In three variables: $\hat{V}_l$, via: $\rho(B)p(x)=p(R^{-1}x)$. Is this just and explicit construction of $\rho(\phi^{-1}B)$, and if so, what is the connection between $V_l$ and $\hat{V}_l$. Thank you:)
Given any irreducible representation $\pi$ of $SO(3)$, the composition $\pi\circ\phi$ is an irreducible representation of $SU(2)$ coinciding on $\pm 1$, so it is necessarily one of the $V_l$, with even $l$.