Torsors on moduli spaces of principal G-bundles on curves

Abstract

Let [Formula: see text] be a semisimple complex algebraic group with a simple Lie algebra [Formula: see text], and let [Formula: see text] denote the moduli stack of topologically trivial stable [Formula: see text]-bundles on a smooth projective curve [Formula: see text]. Fix a theta characteristic [Formula: see text] on [Formula: see text] which is even in case [Formula: see text] is odd. We show that there is a nonempty Zariski open substack [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text], for all [Formula: see text]. It is shown that any such [Formula: see text] has a canonical connection. It is also shown that the tangent bundle [Formula: see text] has a natural splitting, where [Formula: see text] is the restriction of [Formula: see text] to the semi-stable locus. We also produce an isomorphism between two naturally occurring [Formula: see text]-torsors on the moduli space of regularly stable [Formula: see text].