Reductive groups, the loop Grassmannian, and the Springer resolution

Pramod N. Achar; Simon Riche

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arXiv · 1602.04412

Reductive groups, the loop Grassmannian, and the Springer resolution

Pramod N. Achar·Simon Riche

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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-Mirkovi\'c describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian.

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BibTeX RIS

Pramod N. Achar, Simon Riche. 2016-02-14. Reductive groups, the loop Grassmannian, and the Springer resolution. https://arxiv.org/abs/1602.04412

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