Torsors on moduli spaces of principal G-bundles

Let G be a semisimple complex algebraic group with a simple Lie algebra g, and let M⁰G denote the moduli stack of topologically trivial stable G-bundles on a smooth projective curve C. Fix a theta characteristic κ on C which is even in case g is odd. We show that there is a nonempty Zariski open substack U_κ of M⁰G such that Hⁱ(C,\, ad(EG)⊗κ) \,=\, 0, $i\,=\, 1,\, 2$, for all EG\,∈\, U_κ. It is shown that any such EG has a canonical connection. It is also shown that the tangent bundle TU_κ has a natural splitting, where Uκ is the restriction of Uκ to the semi-stable locus. We also produce an isomorphism between two naturally occurring Ω¹_MʳˢG--torsors on the moduli space of regularly stable MʳˢG.