Torsors on moduli spaces of principal G-bundles

Let G be a semisimple complex algebraic group with a simple Lie algebra g, and let M⁰₆ denote the moduli stack of topologically trivial stable G-bundles on a smooth projective curve C. Fix a theta characteristic on C which is even in case g is odd. We show that there is a nonempty Zariski open substack U_ of M⁰₆ such that Hⁱ (C, \, ad (EG) ) \, =\, 0, i\, =\, 1, \, 2, for all EG\, \, U_. It is shown that any such EG has a canonical connection. It is also shown that the tangent bundle TU_ has a natural splitting, where U_ is the restriction of U_ to the semi-stable locus. We also produce an isomorphism between two naturally occurring ¹₌^{rs₆}--torsors on the moduli space of regularly stable M^rs₆.