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 K\"ahler manifold $X$, the dimensional reduction of $G$-equivariant holomorphic vector bundles over $X\times G/P$ was carried out by the first and third authors. This raises the question of dimensional reduction of holomorphic principal bundles over $X\times G/P$. The method used for equivariant vector bundles 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 $X\times G/P$, 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.