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 $({E\prime }_G,{E\prime }_K)$ are called holomorphically isometric if there is a holomorphic isomorphism of the principal $G$-bundle $E_G$ with $E{\prime }_G$ which takes $E_K$ to $E{\prime }_K$. 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 $\chi :\mathbb{R}\to K$ is a homomorphism, and $A\in \mathfrak{k}{\otimes }_{\mathbb{R}}\mathbb{C}$ such that $[A,d\chi (1\left)\right]=2\sqrt{-1}\cdot A$. (Here $d\chi :\mathbb{R}\to \mathfrak{k}$ is the homomorphism of Lie algebras associated to $\chi$.) Two such pairs $(\chi ,A)$ and $(\chi \prime ,A\prime )$ are called equivalent if there is an element $g_0\in K$ such that $\chi \prime =Ad\left(g_0\right)◦\chi$ and $A\prime =Ad\left(g_0\right)\left(A\right)$.