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. This generalizes Simpson's results for GL(n,C) to arbitrary complex and real reductive Lie groups. Three interesting features are the relation between the parabolic degree and the Tits geometry of the boundary at infinity of the symmetric space, 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.