Homogeneous principal bundles over the upper half-plane
Biswas, Indranil
Kyoto J. Math., Tome 50 (2010) no. 2, p. 325-363 / Harvested from Project Euclid
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 $A'=\operatorname{Ad}(g_{0})(A)$ .
Publié le : 2010-05-15
Classification:  53B35,  32L05
@article{1273236819,
     author = {Biswas, Indranil},
     title = {Homogeneous principal bundles over the upper half-plane},
     journal = {Kyoto J. Math.},
     volume = {50},
     number = {2},
     year = {2010},
     pages = { 325-363},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1273236819}
}
Biswas, Indranil. Homogeneous principal bundles over the upper half-plane. Kyoto J. Math., Tome 50 (2010) no. 2, pp.  325-363. http://gdmltest.u-ga.fr/item/1273236819/