Elliptic Springer Theory

TL;DR

This paper develops an elliptic analogue of Springer theory, introducing elliptic sheaves and establishing their properties, which connect geometric Eisenstein series, moduli of G-bundles, and representations of elliptic Weyl groups.

Contribution

It introduces an elliptic version of the Grothendieck-Springer sheaf and establishes foundational results linking geometry and representation theory in the elliptic setting.

Findings

Constructed elliptic Springer sheaves and proved their basic properties.

Embedded elliptic Weyl group representations into perverse sheaves on G-bundle moduli.

Identified elliptic character sheaves as principal series examples.

Abstract

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.