Stable Principal Bundles and Reduction of Structure Group

Let EG be a stable principal G–bundle over a compact connected Kähler manifold, where G is a connected reductive linear algebraic group defined over C. Let H ⊂ G be a complex reductive subgroup which is not necessarily connected, and let EH ⊂ EG be a holomorphic reduction of structure group. We prove that EH is preserved by the Einstein–Hermitian connection on EG. Using this we show that if EH is a minimal reductive reduction in the sense that there is no complex reductive proper subgroup of H to which EH admits a holomorphic reduction of structure group, then EH is unique in the following sense: For any other minimal reductive reduction (H ′ , EH′ ) of EG, there is some element g ∈ G such that H ′ = gHg and EH′ = EHg. As an application, we give an affirmative answer to a question posed in [BK].