[Bezrukavnikov-Mirkovic-Rumynin]-localization for Lie algebras in positive characteristic, and Lusztig's conjectures
by Vinoth Nandakumar
Abstract: This is an expository overview of Bezrukavnikov-Mirkovic-Rumynin's localization theory, which gives derived equivalences between representation categories of Lie algebras in positive characteristic, and coherent sheaves on Springer fibers. The resulting "exotic t-structures" can also be defined geometrically using an action of the affine braid group, and are used to prove Lusztig's conjectures that the classes of the simple modules give a canonical basis in the Grothendieck group of a Springer fiber. For the case of a two-row nilpotent in type A, in joint work with Rina Anno and David Yang, we use [BMR]'s techniques in conjunction with Cautis-Kamnitzer's tangle categorification results to give combinatorial formulae for the dimension of the irreducible modules.
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