Jordan–Hölder reductions for principal Higgs bundles on curves

It is known that semistable sheaves VV admit a filtration whose quotients are stable and have the same slope of VV, named the Jordan–Hölder filtration. We give the analogous result for principal Higgs bundles on curves. Let GG be a reductive algebraic group over CC, if E=(E,ϕ)E=(E,ϕ) is a semistable principal Higgs GG-bundle, there exists a parabolic subgroup PP of GG and an admissible reduction of the structure group of EE to that parabolic such that the principal Higgs bundle obtained by extending the structure group to the Levi factor L(P)L(P) of PP is a stable principal Higgs bundle. The extension of the structure group L(P)L(P) to GG of the latter stable principal bundle is the graded module gr(E).