  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 harmonic
forms $H^1(\Gamma\backslash G/K, \rho)$ are $(0, 1)$-forms for the automorphic
holomorphic bundle induced by the representation $S^m(\mathbb C^{n})$ of $K$.
