Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution
Representations of semisimple Lie algebras in prime characteristic and the noncommutative Springer resolution
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素数特征半单李代数的表示和非交换Springer解析
DOI：
10.4007/annals.2013.178.3.2
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发表时间：
2013
影响因子：
4.9
通讯作者：
I. Mirkovic
中科院分区：
文献类型：
--
作者：
R. Bezrukavnikov;I. Mirkovic
本文在规范基上证明了Lusztig的大部分定理。该图预测，这个基础控制的数字表示的李代数的半单代数群在代数封闭领域的积极特征。我们检查了几乎所有的特征。为此，我们构造了一个非交换决议的幂零锥导出等价于Springer决议。一方面，这个非交换决议是密切相关的积极的特征，本作者和Rumynin早期获得的本地化等价。另一方面，它是兼容的t-结构所产生的一个等价与导范畴的反常层的ane ag簇的朗兰兹对偶群。Arkhipov和第一作者建立的这个等价关系构成了局部几何Langlands对偶的框架。后者的兼容性允许一个应用Frobenius纯度定理推导出所需的性质的基础。我们期望非交换对应的Springer决议是独立的兴趣，从代数几何和几何朗兰兹对偶的角度。
We prove most of Lusztig’s conjectures on the canonical basis in homology of a Springer ber. The conjectures predict that this basis controls numerics of representations of the Lie algebra of a semisimple algebraic group over an algebraically closed eld 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 ane ag variety of the Langlands dual group. This equivalence established by Arkhipov and the rst author ts the framework of local geometric Langlands 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.