Monodromy group for a strongly semistable principal bundle over a curve

Let G be a semisimple linear algebraic group defined over an algebraically closed field k . Fix a smooth projective curve X defined over k , and also fix a closed point x ∈ X . Given any strongly semistable principal G -bundle E G over X , we construct an affine algebraic group scheme defined over k , which we call the monodromy of E G . The monodromy group scheme is a subgroup scheme of the fiber over x of the adjoint bundle E G × G G for E G . We also construct a reduction of structure group of the principal G -bundle E G to its monodromy group scheme. The construction of this reduction of structure group involves a choice of a closed point of E G over x . An application of the monodromy group scheme is given. We prove the existence of strongly stable principal G -bundles with monodromy G