Homogeneous principal bundles over the upper half-plane
Let \(G\) be a connected complex reductive linear algebraic group, \({\mathbb H}\) the upper half-plane. A holomorphic principal \(G\)-bundle over \({\mathbb H}\) is a complex manifold \(E_G\) and a surjective holomorphic submersion \(f:E_G\rightarrow{\mathbb H}\) such that the complex manifold \(E_G\) is equipped with a right holomorphic action of the complex Lie group \(G\), \(\varphi:E_G\times G\rightarrow E_G\), \(f\circ\varphi=f\circ p_1\), where \(p_1\) is the projection of \(E_G\times G\) to the first factor, and the action of \(G\) is free and transitive on each fiber of \(f\). Let \(K\subset G\) be a maximal compact subgroup. The Lie algebra of \(K\) is denoted by \(\mathcal K\). 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 the 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 parametrized by the equivalence classes of pairs of the form \((\chi,A)\), where \(\chi: {\mathbb R}\rightarrow K\) is a homomorphism, and \(A\in{\mathcal K}\otimes_{\mathbb R}{\mathbb C}\) such that \([A,d\chi(1)]=2\sqrt{-1}\cdot A\). (Here \(d\chi:{\mathbb R}\rightarrow{wk}\) 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 \(\chi^\prime = \text{Ad}(g_0)\circ\chi\) and \(A^\prime =\text{Ad}(g_0)(A)\).
scientific article; zbMATH DE number 5735953