2010/01/14 by Roman Bezrukavnikov, Bezrukavnikov, Roman, Ivan Mirković +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1001.2562
openalex publication_date 2010/01/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We prove most of Lusztig's conjectures from the paper "Bases in equivariant\nK-theory II", including the existence of a canonical basis in the Grothendieck\ngroup of a Springer fiber. The conjectures also predict that this basis\ncontrols numerics of representations of the Lie algebra of a semi-simple\nalgebraic group over an algebraically closed field of positive characteristic.\nWe check this for almost all characteristics.\n To this end we construct a non-commutative resolution of the nilpotent cone\nwhich is derived equivalent to the Springer resolution. On the one hand, this\nnoncommutative resolution is shown to be compatible with the positive\ncharacteristic version of Beilinson-Bernstein localization equivalences. On the\nother hand, it is compatible with the t-structure arising from the equivalence\nof Arkhipov-Bezrukavnikov with the derived category of perverse sheaves on the\naffine flag variety of the Langlands dual group, which was inspired by local\ngeometric Langlands duality. This allows one to apply Frobenius purity theorem\nto deduce the desired properties of the basis.\n We expect the noncommutative counterpart of the Springer resolution to be of\nindependent interest from the perspectives of algebraic geometry and geometric\nLanglands duality.\n