Torsors on moduli spaces of principal $G$-bundles

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