Eichler-Shimura isomorphism for complex hyperbolic lattices

We consider the cohomology group H1(Γ,ρ) of a discrete subgroup Γ⊂G=SU(n,1) and the symmetric tensor representation ρ on Sm(Cn+1). We give an elementary proof of the Eichler-Shimura isomorphism that harmonic forms H1(Γ∖G∕K,ρ) are (0,1)-forms for the automorphic holomorphic bundle induced by the representation Sm(Cn) of K.