Given a discrete subgroup of finite co-volume of , we define and study parabolic vector bundles on the quotient of the (extended) hyperbolic plane by . If contains an orientation-reversing isometry, then the above is equivalent to studying real and quaternionic parabolic vector bundles on the orientation cover of degree two of . We then prove that isomorphism classes of polystable real and quaternionic parabolic vector bundles are in a natural bijective correspondence with the equivalence classes of real and quaternionic unitary representations of . Similar results are obtained for compact-type real parabolic vector bundles over Klein surfaces.
"Parabolic vector bundles on Klein surfaces." Illinois J. Math. 64 (1) 105 - 118, April 2020. https://doi.org/10.1215/00192082-8165614