The elliptic Grothendieck-Springer resolution as a simultaneous log resolution of algebraic stacks

Let $G$ be a simply connected simple algebraic group. The classical multiplicative and additive Grothendieck-Springer resolutions are simultaneous resolutions of singularities for the maps from $G$ and its Lie algebra to their invariant theory quotients by the conjugation action of $G$. In this paper, we construct an elliptic version of the Grothendieck-Springer resolution, which is a simultaneous log resolution of a family whose total space is the stack of principal $G$-bundles on an elliptic curve. Our construction extends a well-known simultaneous resolution of the coarse moduli space map for semistable principal bundles to the stack of all principal bundles. We also prove elliptic versions of the Chevalley isomorphism and the Kostant and Steinberg section theorems, on which our construction relies.