Eichler-Shimura isomorphism for complex hyperbolic lattices

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