Elliptic Springer theory

We introduce an elliptic version of the Grothendieck–Springer sheaf and establish elliptic analogues of the basic results of Springer theory. From a geometric perspective, our constructions specialize geometric Eisenstein series to the resolution of degree-zero, semistable $G$ -bundles by degree-zero $B$ -bundles over an elliptic curve $E$ . From a representation theory perspective, they produce a full embedding of representations of the elliptic or double affine Weyl group into perverse sheaves with nilpotent characteristic variety on the moduli of $G$ -bundles over $E$ . The resulting objects are principal series examples of elliptic character sheaves, objects expected to play the role of character sheaves for loop groups.