On the principal bundles with parabolic structure (Q1890280)

Let \(G\) be a semisimple complex algebraic group. For a parabolic principal \(G\)-bundle \(E_*\) on a smooth curve, the author proves the following analogue of Weil's theorem: \(E_*\) admits a flat connection if and only if every direct summand of the adjoint bundle \(E_*({\mathfrak g})\) has parabolic degree zero (\({\mathfrak g}\) being the Lie algebra of \(G\)). Given a polynomial \(P(x)\) with non-negative integral coefficients and a parabolic vector bundle \(F_*\), define \(P(F_*)\) to be the parabolic vector bundle obtained by putting \(F_*\) for \(x\) and replacing addition and multiplication by direct sum and tensor product respectively. Then \(F_*\) is called finite if there exist two such distinct polynomials \(P_1\) and \(P_2\) with \(P_1(F_*) \cong P_2(F_*)\). A parabolic principal \(G\)-bundle \(E_*\) on a projective manifold \(X\) is called finite if all the parabolic vector bundles associated to \(E_*\) via representations are finite. The author shows that \(E_*\) is finite if and only if it admits a flat connection with a finite monodromy group.