Homogeneous principal bundles over the upper half-plane

Abstract

Let $G$ be a connected complex reductive linear algebraic group, and let $K\subset G$ be a maximal compact subgroup. The Lie algebra of $K$ is denoted by $\mathfrak{k}$. A holomorphic Hermitian principal $G$-bundle is a pair of the form $(E_{G},E_{K})$, where $E_{G}$ is a holomorphic principal $G$-bundle and $E_{K}\subset E_{G}$ is a $C^{\infty}$-reduction of structure group to $K$. Two holomorphic Hermitian principal $G$-bundles $(E_{G},E_{K})$ and $(E'_{G},E'_{K})$ are called holomorphically isometric if there is a holomorphic isomorphism of the principal $G$-bundle $E_{G}$ with $E'_{G}$ which takes $E_{K}$ to $E'_{K}$. We consider all holomorphic Hermitian principal $G$-bundles $(E_{G},E_{K})$ over the upper half-plane $\mathbb{H}$ such that the pullback of $(E_{G},E_{K})$ by each holomorphic automorphism of $\mathbb{H}$ is holomorphically isometric to $(E_{G},E_{K})$ itself. We prove that the isomorphism classes of such pairs are parameterized by the equivalence classes of pairs of the form $(\chi ,A)$, where $\chi \dvtx {\mathbb{R}}\longrightarrow K$ is a homomorphism, and $A\in \mathfrak{k}\otimes_{\mathbb{R}}{\mathbb{C}}$ such that $[A,d\chi(1)]=2\sqrt{-1}{\cdot}A$. (Here $d\chi \dvtx{\mathbb{R}}\longrightarrow \mathfrak{k}$ is the homomorphism of Lie algebras associated to $\chi$.) Two such pairs $(\chi ,A)$ and $(\chi',A')$ are called equivalent if there is an element $g_{0}\in K$ such that $\chi'=\operatorname{Ad}(g_{0})\circ\chi$ and $A'=\operatorname{Ad}(g_{0})(A)$.