Parabolic vector bundles on Klein surfaces

Abstract

Given a discrete subgroup Γ of finite co-volume of PGL(2,R), we define and study parabolic vector bundles on the quotient Σ of the (extended) hyperbolic plane by Γ. If Γ 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 Σ. 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 Γ. Similar results are obtained for compact-type real parabolic vector bundles over Klein surfaces.