A criterion for homogeneous principal bundles

TL;DR

This paper establishes criteria for when principal bundles over homogeneous spaces are homogeneous, linking the homogeneity of the bundle to the homogeneity of its adjoint and associated vector bundles.

Contribution

It provides new conditions under which a principal bundle's homogeneity can be deduced from the homogeneity of related vector bundles.

Findings

A holomorphic principal H-bundle is homogeneous if its adjoint bundle is homogeneous.

Homogeneity of the bundle can be inferred from the homogeneity of associated vector bundles for any faithful H-module.

The results apply to principal bundles over spaces G/P where P is a parabolic subgroup of a semisimple group.

Abstract

We consider principal bundles over homogeneous spaces G/P, where P is a parabolic subgroup of a semisimple and simply connected complex linear algebraic group G. We prove that a holomorphic principal H--bundle, where H is a complex reductive group, is homogeneous if the adjoint vector bundle ad(E) is homogeneous. We also show that E is homogeneous if its associated vector bundle for any finite dimensional faithful H--module is homogeneous.