Torsors on loop groups and the Hitchin fibration

TL;DR

This paper extends the product formula for the Hitchin fibration to a larger locus by analyzing torsors over loop groups, providing new proofs and geometric insights related to algebraization and invariance properties.

Contribution

It establishes the product formula over the generically regular semisimple locus and introduces new algebraization, approximation, and invariance results for torsors, along with improved geometric understanding.

Findings

Confirmed the product formula over the larger locus

Derived vanishing statements for torsors over loop groups

Provided new proofs for key algebraic and geometric theorems

Abstract

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 $\mathfrak{g} \rightarrow \mathfrak{g}//G$.