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Let $P$ be a parabolic subgroup of a connected simply connected complexsemisimple Lie group $G$. Given a compact K\"ahler manifold $X$, thedimensional reduction of $G$-equivariant holomorphic vector bundles over$X\times G/P$ was carried out by the first and third authors. This raises thequestion of dimensional reduction of holomorphic principal bundles over$X\times G/P$. The method used for equivariant vector bundles does notgeneralize to principal bundles. In this paper, we adapt to equivariantprincipal bundles the Tannakian approach of Nori, to describe the dimensionalreduction of $G$-equivariant principal bundles over $X\times G/P$, and toestablish a Hitchin--Kobayashi type correspondence. In order to be able toapply the Tannakian theory, we need to assume that $X$ is a complex projectivemanifold.

A Tannakian approach to dimensional reduction of principal bundles