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Let $G$ be a simply connected simple algebraic group. The classicalmultiplicative and additive Grothendieck-Springer resolutions are simultaneousresolutions of singularities for the maps from $G$ and its Lie algebra to theirinvariant theory quotients by the conjugation action of $G$. In this paper, weconstruct an elliptic version of the Grothendieck-Springer resolution, which isa simultaneous log resolution of a family whose total space is the stack ofprincipal $G$-bundles on an elliptic curve. Our construction extends awell-known simultaneous resolution of the coarse moduli space map forsemistable principal bundles to the stack of all principal bundles. We alsoprove elliptic versions of the Chevalley isomorphism and the Kostant andSteinberg section theorems, on which our construction relies.

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