On the stable principal Higgs sheaves

Let G   be a connected reductive linear algebraic group defined over CC with Lie algebra gg. Let (EG,E,ψ,φ,φˆ) be a stable principal Higgs G  -sheaf on a compact connected Kähler manifold. We consider all holomorphic sections of the adjoint vector bundle ad(EG)ad(EG) of EGEG that commute with the Higgs field φ. These correspond to the infinitesimal automorphisms of the principal Higgs G  -sheaf. Any element of the center of gg gives such a section. We prove that all the sections are given by the center of gg.