Reductive groups, the loop Grassmannian, and the Springer resolution

• 成果类型: Article
• 署名作者: Achar, Pramod N.; Riche, Simon
• 署名单位: Louisiana State University System; Louisiana State University; Universite Clermont Auvergne (UCA); Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-018-0805-1
• 发表日期: 2018
• 页码: 289-436
• 关键词: kazhdan-lusztig conjecture; affine lie-algebras; koszul duality; modular-representations; intersection cohomology; inversion-formula; quantum groups; localization; sheaves; POLYNOMIALS

摘要

In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a graded version of a conjecture of Finkelberg-Mirkovi describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian, and deduce a new proof of Lusztig's conjecture in large characteristic.