On the vector bundles associated to irreducible representations of cocompact lattices of SL(2,C)

In this continuation of \cite{BM}, we prove the following: Let $\Gamma\subset \text{SL}(2,{\mathbb C})$ be a cocompact lattice, and let $\rho: \Gamma \rightarrow \text{GL}(r,{\mathbb C})$ be an irreducible representation. Then the holomorphic vector bundle $E_\rho \longrightarrow \text{SL}(2,{\mathbb C})/\Gamma$ associated to $\rho$ is polystable. The compact complex manifold $\text{SL}(2,{\mathbb C})/\Gamma$ has natural Hermitian structures; the polystability of $E_\rho$ is with respect to these natural Hermitian structures.