Advances in Theoretical and Mathematical Physics

Volume 17 (2013)

Number 6

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

Pages: 1417 – 1424



Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Bombay, India)

Avijit Mukherjee (Department of Physics, Jadavpur University, Jadavpur, Kolkata, India)


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.

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