A Higgs bundle on a Hermitian symmetric space

Let (M := Gammabackslash G/K) be the quotient of an irreducible Hermitian symmetric space G/K by a torsionfree cocompact lattice (Gammasubset G) . There is a natural flat principal G-bundle over the compact Kähler manifold M which is constructed from the principal Γ-bundle over M defined by the quotient map (G/Klongrightarrow M) . We construct the principal G-Higgs bundle over M corresponding to this flat G-bundle. This principal G-Higgs bundle is rigid if ({rm dim}_mathbb{C} M,geq,2) .