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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–Mirković 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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Inventiones Mathematicae

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