On holomorphic principal bundles over a compact Riemann surface admitting a flat connection

Let G be a connected reductive linear algebraic group over C, and X a compact connected Riemann surface. Let L subset of G be a Levi factor of some parabolic subgroup of G, with L-0 = L/[L, L] its maximal abelian quotient. We prove that a holomorphic G-bundle E-G over X admits a flat connection if and only if for every such L and every reduction E-L subset of E-G Of the structure group of E-G to L, the L-0-bundle obtained by extending the structure group of E-L is topologically trivial. For G = GL(n, C) this is a well-known result of A. Weil.