= Homogeneous line-bundle representations on the two-sphere
{title2=$L^2(\mathcal O(q))=\widehat\bigoplus_{n\geq|q|,\,n\equiv q\ (2)}\operatorname{Sym}^n\mathbb C^2$}
On $S^2=SU(2)/U(1)$ a degree-$q$ equivariant line bundle corresponds to isotropy character $\operatorname{diag}(e^{it},e^{-it})\mapsto e^{-iqt}$. The <Peter-Weyl theorem> computes its section multiplicities by the occurrence of weight $-q$ in the <SU(2) representation> $\operatorname{Sym}^n\mathbb C^2$. Its weights $n,n-2,\ldots,-n$ give the displayed multiplicity-one decomposition. The spinor lines have degrees $-1,+1$ and thus contain all odd $n$, once each.
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