Highest weight modules at the critical level and noncommutative Springer resolution
Author(s)Bezrukavnikov, Roman; Lin, Qian
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In the article by Bezrukavnikov and Mirkovic 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 gˇ be the Langlands dual Lie algebra and let [˄ over g] be the corresponding affine Lie algebra, i.e. [˄ over g] is a central extension of gˇ ⊗ C((t)). Using results of Frenkel and Gaitsgory we show that the category of [˄ over 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 central reduction of A. 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 by Bezrukavnikov and Mirkovic.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Algebraic Groups and Quantum Groups
American Mathematical Society
Bezrukavnikov, Roman, and Qian Lin. Highest weight modules at the critical level and noncommutative Springer resolution. American Mathematical Society, 2012.
Author's final manuscript