On the vector bundles associated to the irreducible representations of cocompact lattices of \(\mathrm{SL}(2,\mathbb{C})\). (English) Zbl 1300.32020

Biswas, Indranil; Mukherjee, Avijit

Adv. Theor. Math. Phys. 17, No. 6, 1417-1424 (2013).

Summary: We prove the following: let \(\gamma\subset \mathrm{SL}(2,\mathbb C)\) be a cocompact lattice and let \(\rho:\Gamma\to \mathrm{GL}(2,\mathbb C)\) be an irreducible representation. Then the holomorphic vector bundle \(E_\rho\to \mathrm{SL}(2,\mathbb C)/\Gamma\) associated to \(\rho\) is polystable. The compact complex manifold \(\mathrm{SL}(2,\mathbb C)/\Gamma\) 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.

MSC:

32L05	Holomorphic bundles and generalizations
32M05	Complex Lie groups, group actions on complex spaces

Keywords:

holomorphic vector bundles; irreducible representations of cocompact lattices in \(\mathrm{SL}(2,\mathbb C)\)

Citations:

Zbl 1272.32021