Eichler-Shimura isomorphism for complex hyperbolic lattices

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

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