Harder-Narasimhan reduction of a principal bundle

Abstract

Let $E$ be a principal $G$--bundle over a smooth projective curve over an algebraically closed field $k$, where $G$ is a reductive linear algebraic group over $k$. We construct a canonical reduction of $E$. The uniqueness of canonical reduction is proved under the assumption that the characteristic of $k$ is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of $k$ is positive.