Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution. (English) Zbl 1293.17021

Bezrukavnikov, Roman; Mirković, Ivan

Ann. Math. (2) 178, No. 3, 835-919 (2013).

In [Represent. Theory 3, 281–353 (1999; Zbl 0999.20036)] G. Lusztig conjectured the existence of a certain basis in the Grothendieck group of a Springer fiber and the fact that this basis controls numerical invariants of irreducible finite dimensional representations of semisimple Lie algebras in positive characteristic. In the paper under review the authors prove this in almost all cases (that is for all sufficiently large characteristics). The first step of the proof involves construction of a certain noncommutative algebra (a kind of a noncommutative counterpart for the Springer resolution) which lifts modular representation categories to characteristic zero. This algebra turns out to be determined by a certain \(t\)-structure on the bounded derived category of coherent sheaves on the cotangent bundle of the flag variety. Checking of one of Lusztig’s axioms reduces to a certain compatibility property between the \(t\)-structure mentioned above and the multiplicative group action on Slodowy slices and, basically, says that the grading on the slice algebra is positive, which is proved using an appropriate variant of the purity theorem.

Reviewer: Volodymyr Mazorchuk (Uppsala)

MSC:

17B50	Modular Lie (super)algebras
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
14L35	Classical groups (algebro-geometric aspects)

Keywords:

affine flag variety; canonical basis; representation; Lie algebra; noncommutative Springer resolution

Citations:

Zbl 0999.20036

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