A criterion for flatness of sections of adjoint bundle of a holomorphic principal bundle over a Riemann surface (Q1946064)

Let \(X\) be a compact connected Riemann surface and \(G\) a connected reductive affine group defined over \(\mathbb C\). Let \(E_G\) be a holomorphic principal \(G\)-bundle over \(X\). Take \(\text{ad}(E_G):=E_G \times ^G \mathfrak g\), the adjoint vector bundle, where \(\mathfrak g\) is the Lie algebra of \(G\). In this paper one gives sufficient conditions for a section \(\beta\) in \(H^0(X, \text{ad}(E_G))\), which has the property that all \(\beta (x)\) (\(x \in X\)) have the same image in \(\mathfrak{g}/G: =\) \{ {\text{ the set of all conjugacy classes in}} \(\mathfrak g\) \}, to have the property that there exists a holomorphic connection on \(E_G\) such that \(\beta\) is flat with respect to the connection induced on \(\text{ad}(E_G)\). The condition is given, among others, in terms of a Levi subgroup of a parabolic subgroup of \(G\), defined naturally as the centralizer of an element in \(\mathfrak g\) whose image in \(\mathfrak{g}/G\) coincide with the image of \(\beta _s(x)\), for \(\beta _s\) a semisimple section. An interesting form for \(G=\mathrm{GL}(r,\mathbb C)\) is given.