Homogeneous line-bundle representations on the two-sphere ($L^2(\mathcal{O}(q))={\widehat{⨁}}_{n\ge |q|,n\equiv q(2)}{\mathrm{Sym}}^n{\mathbb{C}}^2$)

On $S^2=SU(2)/U(1)$ a degree-$q$ equivariant line bundle corresponds to isotropy character $\mathrm{diag}(e^{it},e^{-it})\mapsto e^{-iqt}$. The Peter-Weyl theorem computes its section multiplicities by the occurrence of weight $-q$ in the SU(2) representation ${\mathrm{Sym}}^n{\mathbb{C}}^2$. Its weights $n,n-2,\dots ,-n$ give the displayed multiplicity-one decomposition. The spinor lines have degrees $-1,+1$ and thus contain all odd $n$, once each.