Equivariant principal bundles over the complex projective line

Abstract

Let $G$ be a connected complex reductive linear algebraic group, and let $K \subset G$ be a maximal compact subgroup of it. Let $E_G$ be a holomorphic principal $G$-bundles over the complex projective line ${\mathbb C}{\mathbb P}^1$ and $E_K \subset E_G$ a $C^\infty$ reduction of structure group of $E_G$ to $K$. We consider all pairs $(E_G ,E_K)$ of this type such that the total space of $E_K$ is equipped with a $C^\infty$ lift of the standard action of $\operatorname{SU}(2)$ on ${\mathbb C}{\mathbb P}^1$ which satisfies the following two conditions: the actions of $K$ and $\operatorname{SU}(2)$ on $E_K$ commute, and for each element $g \in \operatorname{SU}(2)$, the induced action of $g$ on $E_G$ is holomorphic. We give a classification of the isomorphism classes of all such objects.