Parabolic Higgs bundles and representations of the fundamental group of a punctured surface into a real group

Abstract

We study parabolic G-Higgs bundles over a compact Riemann surface with fixed punctures, when G is a real reductive Lie group, and establish a correspondence between these objects and representations of the fundamental group of the punctured surface in G with arbitrary holonomy around the punctures. Three interesting features are the relation between the parabolic degree and the geometry of the Tits boundary, the treatment of the case when the logarithm of the monodromy is on the boundary of a Weyl alcove, and the correspondence of the orbits encoding the singularity via the Kostant-Sekiguchi correspondence. We also describe some special features of the moduli spaces when G is a split real form or a group of Hermitian type.

arXiv commentv2: references added, a few typos corrected; v3: substantial revision, added three sections: (3) relation to parahoric bundles, (7) moduli spaces, (8) examples, 60 pages; v4: several corrections, more details included on the Hitchin-Kobayashi correspondence, some proofs shortened, 57 pages