Torsors on loop groups and the Hitchin fibration

In his proof of the fundamental lemma, Ng\^o established the product formula for the Hitchin fibration over the anisotropic locus. One expects this formula over the larger generically regular semisimple locus, and we confirm this by deducing the relevant vanishing statement for torsors over loop groups $R((t))$ from a general formula for $\mathrm{Pic}(R((t)))$. In the build up to the product formula, we present general algebraization, approximation, and invariance under Henselian pairs results for torsors, give short new proofs for the Elkik approximation theorem and the Chevalley isomorphism $\mathfrak{g}//G \cong \mathfrak{t}/W$, and improve results on the geometry of the Chevalley morphism $\mathfrak{g} \rightarrow \mathfrak{g}//G$.