On the equivariant vector bundles on ${\mathbb{C}\mathbb{P}}^1$

Let $H$ be a subgroup of $\mathrm{P}\mathrm{G}\mathrm{L}(2,\mathbb{C})$ (respectively, $\mathrm{S}\mathrm{L}(2,\mathbb{C})$) such that the Zariski closure in $\mathrm{P}\mathrm{G}\mathrm{L}(2,\mathbb{C})$ (respectively, $\mathrm{S}\mathrm{L}(2,\mathbb{C})$) of some compact subgroup of $H$ contains $H$. We classify the $H$–equivariant holomorphic vector bundles on ${\mathbb{C}\mathbb{P}}^1$. This generalizes \cite{BM} where $H$ was assumed to be a finite abelian group.