Highest weight modules at the critical level and noncommutative Springer resolution

TL;DR

This paper establishes a deep connection between categories of affine Lie algebra modules at the critical level and modules over a non-commutative algebra related to Springer resolution, revealing new insights into their structure and numerical properties.

Contribution

It proves an equivalence between categories of affine Lie algebra modules at the critical level and modules over a quotient of a non-commutative algebra linked to Springer resolution, confirming conjectures about their structure.

Findings

Category of affine Lie algebra modules at critical level is equivalent to modules over a quotient of algebra A.

Numerics of Iwahori integrable modules are governed by the canonical basis in the K-group of Springer fiber.

Supports conjectural descriptions of module structures by Lusztig.

Abstract

In arXiv:1001.2562 a certain non-commutative algebra $A$ was defined starting from a semi-simple algebraic group, so that the derived category of $A$-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution. Let $\hat{\g}$ be the affine Lie algebra corresponding to the Langlands dual Lie algebra. Using results of Frenkel and Gaitsgory arXiv:0712.0788 we show that the category of $\hat{\g}$ modules at the critical level which are Iwahori integrable and have a fixed central character, is equivalent to the category of modules over a quotient of $A$ by a central character. This implies that numerics of Iwahori integrable modules at the critical level is governed by the canonical basis in the $K$-group of a Springer fiber, which was conjecturally described by Lusztig and constructed in arXiv:1001.2562.