A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface

Abstract

Let $E_G$ be a holomorphic principal $G$--bundle over a compact connected Riemann surface, where $G$ is a connected reductive affine algebraic group defined over $\ C$, such that $E_G$ admits a holomorphic connection. Take any $\beta \in H^0(X, {ad}(E_G))$, where ${ad}(E_G)$ is the adjoint vector bundle for $E_G$, such that the conjugacy class $\beta (x) \in {\mathfrak g}/G, x \in X$, is independent of $x$. We give a sufficient condition for the existence of a holomorphic connection on $E_G$ such that $\beta$ is flat with respect to the induced connection on ${ad}(E_G)$.