Monodromy group for a strongly semistable principal bundle over a curve

Abstract

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 xεX. 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×^GG 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.