On the stability of flat complex vector bundles over parallelizable manifolds

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G\mathrm{/}\mathrm{\Gamma }$, where $G$ is a complex connected Lie group and $\mathrm{\Gamma }$ is a cocompact lattice in it. The main result proved here is a structure theorem for flat holomorphic vector bundles $E_{\rho }$ associated to any irreducible representation $\rho : \Gamma \rightarrow $\text{GL}$(r,{\mathbb C})$\mathrm{.}Moreprecisely,weprovethat$E_{\rho}$ is holomorphically isomorphic to a vector bundle of the form $E^{\oplus n}$, where $E$ is a stable vector bundle. All the rational Chern classes of $E$ vanish, in particular, its degree is zero. We deduce a stability result for flat holomorphic vector bundles $E_{\rho }$ of rank 2 over $G\mathrm{/}\mathrm{\Gamma }$. If an irreducible representation $\rho : \Gamma\rightarrow $\text{GL}$(2, \mathbb {C})$ satisfies the conditionmthat the induced homomorphism $\mathrm{\Gamma }\to \mathrm{P}\mathrm{G}\mathrm{L}(2,\mathbb{C})$ does not extend to a homomorphism from $G$, then $E_{\rho }$ is proved to be stable.