Let \(G\) be a connected reductive linear algebraic group over \(\mathbb{C}, {\mathfrak g}\) the Lie algebra of \(G\), and \(\Gamma \subset G\) a cocompact discrete subgroup. The Dolbeault cohomology space \(H^{p,q} (G/ \Gamma)\) is considered with the induced representation of \(G\). It is proved that \(H^{p,q} (G/ \Gamma)\) is isomorphic to the tensor product \(H^q (\Gamma, \mathbb{C}) \otimes \wedge^p ({\mathfrak g})\), where \(G\) acts trivially on the first factor and where the action on the second one is induced by the adjoint representation on \({\mathfrak g}\). As a consequence, a new proof of Raghunathan's vanishing theorem for one-dimensional cohomology is obtained. More generally, let \(H\subset G\) be any closed complex Lie subgroup such that \(X=G/H\) is compact, \(H^0 \subset H\) be the connected component of the identity element and \(P\) the normalizer of \(H^0\) in \(G\). It is shown that there exists a connected reductive algebraic subgroup \(G^* \subset P\), such that \(P=G^* \cdot H^0\) and \(\Gamma^* =G^* \cap H\) is discrete. For each \(k\), \(k=0,1, \dots, \dim X\), a spectral sequence of \(G\)-modules is constructed, so that \(E_2^{p,q} =E_2^{p,q} \{k\}= H^p(G/P, \wedge^k ({\mathfrak g}/{\mathfrak h})^*) \otimes H^q (\Gamma^*, \mathbb{C})\), where \(H^* (\Gamma^*, \mathbb{C})\) is as a trivial \(G\)-module. The spectral sequence converges to \(H^*(X,\Omega_X^k)\). As a corollary, it is shown that the \(G\)-action on \(H^q(G/H, {\mathcal O})\) is trivial for all \(q\). The situation changes drastically if \(G\) is non-reductive. As an example, the Dolbeault cohomology \(G\)-modules are computed for homogeneous manifolds \(G/ \Gamma\), where \(\Gamma\) is a certain discrete cocompact subgroup of a semidirect product \(G= (\mathbb{C}^*)^{r-1} \ltimes \mathbb{C}^r\). In particular, it turns out that the representation in \(H^{0,q} (G/ \Gamma)\) is nontrivial. This result is a generalization of an example due to \textit{F. Lescure} [C. R. Acad. Sci. Paris, Sér. I 316, No. 8, 823-825 (1993; Zbl 0784.32011)].