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 $\beta \in H^0(X,\mathrm{ad}(E_G))$, where $\mathrm{ad}(E_G)$ 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 $\mathrm{ad}(E_G)$.