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

We prove most of Lusztig's conjectures on the canonical basis in homology of a Springer fiber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of semisimple algebraic group over an algebraically closed field of positive characteristic. We check th...

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Bibliographic Details
Published in:Annals of mathematics 2013-11, Vol.178 (3), p.835-919
Main Authors: Bezrukavnikov, Roman, Mirković, Ivan
Format: Article
Language:English
Subjects:
Algebra
Braiding
Functors
Geometric shapes
Mathematical theorems
Mathematical vectors
Morphisms
Subalgebras
Uniqueness
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Summary:We prove most of Lusztig's conjectures on the canonical basis in homology of a Springer fiber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of semisimple algebraic group over an algebraically closed field of positive characteristic. We check this for almost all characteristics. To this end we construct a noncommutative resolution of the nilpotent cone which is derived equivalent to the Springer resolution. On the one hand, this noncommutative resolution is closely related to the positive characteristic derived localization equivalences obtained earlier by the present authors and Rumynin. On the other hand, it is compatible with the t-structure arising from an equivalence with the derived category of perverse sheaves on the affine flag variety of the Langlands dual group. This equivalence established by Arkhipov and the first author fits the framework of local geometric Landlands duality. The latter compatibility 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.
ISSN:0003-486X
DOI:10.4007/annals.2013.178.3.2