Advances in Theoretical and Mathematical Physics

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

Indranil Biswas and Avijit Mukherjee

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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.

Article information

Source
Adv. Theor. Math. Phys., Volume 17, Number 6 (2013), 1417-1424.

Dates
First available in Project Euclid: 21 August 2014

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1408626547

Mathematical Reviews number (MathSciNet)
MR3262527

Zentralblatt MATH identifier
1300.32020

Citation

Biswas, Indranil; Mukherjee, Avijit. On the vector bundles associated to the irreducible representations of cocompact lattices of $\text{SL}(2,{\mathbb C})$. Adv. Theor. Math. Phys. 17 (2013), no. 6, 1417--1424. https://projecteuclid.org/euclid.atmp/1408626547


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