Hermitian symmetric space, flat bundle and holomorphicity criterion

Let $K\mathrm{\backslash }G$ be an irreducible Hermitian symmetric space of noncompact type and $\mathrm{\Gamma }\subset G$ a closed torsionfree discrete subgroup. Let $X$ be a compact K"ahler manifold and $\rho, :, \pi_1(X, x_0),\longrightarrow, \Gamma$ahomomorphismsuchthattheadjointactionof$\rho(\pi_1(X, x_0))$on$$\text{Lie}$(G)$iscompletelyreducible\mathrm{.}AtheoremofCorletteassociatesto$\rho$aharmonicmap$X, \longrightarrow, K\backslash G/\Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti–holomorphic.