A Tannakian approach to dimensional reduction of principal bundles

Let P be a parabolic subgroup of a connected simply connected complex semisimple Lie group G. Given a compact Kähler manifold X, the dimensional reduction of G-equivariant holomorphic vector bundles over X×G/P was carried out in Álvarez-Cónsul and García-Prada (2003). This raises the question of dimensional reduction of holomorphic principal bundles over X×G/P. The method of Álvarez-Cónsul and García-Prada (2003) is special to vector bundles; it does not generalize to principal bundles. In this paper, we adapt to equivariant principal bundles the Tannakian approach of Nori, to describe the dimensional reduction of G-equivariant principal bundles over X×G/P, and to establish a Hitchin-Kobayashi type correspondence. In order to be able to apply the Tannakian theory, we need to assume that X is a complex projective manifold.