Logarithmic connections on principal bundles over a Riemann surface

Abstract

Let [Formula: see text] be a holomorphic principal [Formula: see text]-bundle on a compact connected Riemann surface [Formula: see text], where [Formula: see text] is a connected reductive complex affine algebraic group. Fix a finite subset [Formula: see text], and for each [Formula: see text] fix [Formula: see text]. Let [Formula: see text] be a maximal torus in the group of all holomorphic automorphisms of [Formula: see text]. We give a necessary and sufficient condition for the existence of a [Formula: see text]-invariant logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text]. We also give a necessary and sufficient condition for the existence of a logarithmic connection on [Formula: see text] singular over [Formula: see text] such that the residue over each [Formula: see text] is [Formula: see text], under the assumption that each [Formula: see text] is [Formula: see text]-rigid.