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Journal Articles Inventiones Mathematicae Year : 2018

Reductive groups, the loop Grassmannian, and the Springer resolution

Abstract

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-Mirkovic 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.
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Dates and versions

hal-01273980 , version 1 (15-02-2016)
hal-01273980 , version 2 (23-02-2017)
hal-01273980 , version 3 (17-04-2018)

Identifiers

  • HAL Id : hal-01273980 , version 3

Cite

Pramod N. Achar, Simon Riche. Reductive groups, the loop Grassmannian, and the Springer resolution. Inventiones Mathematicae, 2018, 214, pp.289-436. ⟨hal-01273980v3⟩
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