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Let $K\backslash G$ be an irreducible Hermitian symmetric space of noncompacttype and $\Gamma \,\subset\, G$ a closed torsionfree discrete subgroup. Let $X$be a compact K\"ahler manifold and $\rho\, :\, \pi_1(X, x_0)\,\longrightarrow\,\Gamma$ a homomorphism such that the adjoint action of $\rho(\pi_1(X, x_0))$ on$\text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to$\rho$ a harmonic map $X\, \longrightarrow\, K\backslash G/\Gamma$. We give acriterion for this harmonic map to be holomorphic. We also give a criterion forit to be anti--holomorphic.

Hermitian symmetric space, flat bundle and holomorphicity criterion