Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution

Details Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution Representations of semisimple Lie algebras in prime characteristic and noncommutative Springer resolution

2010-01-14

arxiv.org/abs/1001.2562v6

Mathematics

Representation Theory

minor changes compared to the previous version, a "forward reference" has now become an actual reference to arxiv:1101.3702

Scientific paper

We prove most of Lusztig's conjectures from the paper "Bases in equivariant K-theory II", including the existence of a canonical basis in the Grothendieck group of a Springer fiber. The conjectures also predict that this basis controls numerics of representations of the Lie algebra of a semi-simple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a non-commutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is shown to be compatible with the positive characteristic version of Beilinson-Bernstein localization equivalences. On the other hand, it is compatible with the t-structure arising from the equivalence of Arkhipov-Bezrukavnikov with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group, which was inspired by local geometric Langlands duality. This allows one to apply Frobenius purity theorem to deduce the desired properties of the basis. We expect the noncommutative counterpart of the Springer resolution to be of independent interest from the perspectives of algebraic geometry and geometric Langlands duality.

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