On the vector bundles associated to the irreducible representations of cocompact lattices of $\text{SL}(2,{\mathbb C})$

Abstract

We prove the following: let $\Gamma\, \subset\, \text{SL}(2,{\mathbb C})$ be a cocompact lattice and let $\rho\,:\, \Gamma\, \longrightarrow\, \text{GL}(r,{\mathbb C})$ 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. We show that the polystable vector bundle $E_\rho$ is not stable in general.