Reductive groups, the loop Grassmannian, and the Springer resolution

Pramod N. Achar (1) , Simon Riche (2, 3)

1 LSU Mathematics - Department of Mathematics [Baton Rouge]
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]

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.

Domaines: Mathématiques [math]

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