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 $x\in X$. 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 $E_G\times {}^{G}G$ 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 $G$