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

TL;DR

This paper proves that for cocompact lattices in SL(2,C), the vector bundles associated to irreducible representations are polystable with respect to natural Hermitian structures on the quotient manifold.

Contribution

It establishes the polystability of vector bundles linked to irreducible representations over quotients of SL(2,C), extending previous work and providing new geometric stability results.

Findings

Vector bundles are polystable on SL(2,C)/Γ

Polystability holds with respect to natural Hermitian structures

Extends stability results to cocompact lattices in SL(2,C)

Abstract

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.