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We consider the cohomology group $H^1(\Gamma, \rho)$ of a discrete subgroup$\Gamma\subset G=SU(n, 1)$ and the symmetric tensor representation $\rho$ on$S^m(\mathbb C^{n+1})$.  We give an elementary proof of the Eichler-Shimura isomorphism that harmonicforms $H^1(\Gamma\backslash G/K, \rho)$ are $(0, 1)$-forms for the automorphicholomorphic bundle induced by the representation $S^m(\mathbb C^{n})$ of $K$.

Eichler-Shimura isomorphism for complex hyperbolic lattices