Highest weight modules at the critical level and noncommutative Springer resolution

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.