Hermitian symmetric space, flat bundle and holomorphicity criterion

Let K\GK\G be an irreducible Hermitian symmetric space of noncompact type and Γ⊂G a closed torsionfree discrete subgroup. Let X   be a compact Kähler manifold and ρ:π1(X,x0)⟶Γ a homomorphism such that the adjoint action of ρ(π1(X,x0))ρ(π1(X,x0)) on the Lie algebra Lie(G)Lie(G) is completely reducible. A theorem of Corlette associates to ρ   a harmonic map H:X⟶K\G/Γ. We give a criterion for this harmonic map H to be holomorphic. We also give a criterion for it to be anti-holomorphic.