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 $\mathcal{M}^0_{G}$ 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 $\mathcal{M}^0_{G}$ 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 $\Omega^1_{{M}^{rs}_{G}}$--torsors on the moduli space of regularly stable ${M}^{rs}_{G}$.