Parabolic vector bundles on Klein surfaces

TL;DR

This paper studies parabolic vector bundles on hyperbolic surfaces and Klein surfaces, establishing a correspondence between their polystability classes and certain unitary representations, extending classical results to real and quaternionic cases.

Contribution

It introduces a framework for real and quaternionic parabolic vector bundles on Klein surfaces and proves a bijective correspondence with specific unitary representations, extending known theories.

Findings

Polystable real and quaternionic parabolic bundles correspond to real and quaternionic unitary representations.

Established equivalence between parabolic bundles on Klein surfaces and certain representation classes.

Extended classical results to include real and quaternionic structures on hyperbolic and Klein surfaces.

Abstract

Given a discrete subgroup $\Gamma$ of finite co-volume of $\mathrm{PGL}(2,\mathbb{R})$, we define and study parabolic vector bundles on the quotient $\Sigma$ of the (extended) hyperbolic plane by $\Gamma$. If $\Gamma$ contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vector bundles on the orientation cover of $\Sigma$. We then prove that isomorphism classes of polystable real and quaternionic parabolic vector bundles are in bijective correspondence with equivalence classes of real and quaternionic unitary representations of $\Gamma$. Similar results are obtained for compact-type real parabolic vector bundles over Klein surfaces.