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In arXiv:1001.2562 a certain non-commutative algebra $A$ was defined startingfrom a semi-simple algebraic group, so that the derived category of $A$-modulesis equivalent to the derived category of coherent sheaves on the Springer (orGrothendieck-Springer) resolution.  Let $\hat{\g}$ be the affine Lie algebra corresponding to the Langlands dualLie algebra. Using results of Frenkel and Gaitsgory arXiv:0712.0788 we showthat the category of $\hat{\g}$ modules at the critical level which are Iwahoriintegrable and have a fixed central character, is equivalent to the category ofmodules over a quotient of $A$ by a central character. This implies thatnumerics of Iwahori integrable modules at the critical level is governed by thecanonical basis in the $K$-group of a Springer fiber, which was conjecturallydescribed by Lusztig and constructed in arXiv:1001.2562.

Highest weight modules at the critical level and noncommutative Springer resolution