Torsors on Moduli Spaces of Principal G-Bundles on Curves

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.