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Mixed motives and geometric representation theory in equal characteristic

  • Jens Niklas EberhardtEmail author
  • Shane Kelly
Article
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Abstract

Let \(\mathbb {k}\) be a field of characteristic p. We introduce a formalism of mixed sheaves with coefficients in \(\mathbb {k}\) and apply it in representation theory. We construct a system of \(\mathbb {k}\)-linear triangulated category of motives on schemes over \(\overline{\mathbb {F}}_p\), which has a six functor formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of mixed sheaves that one is used to. We attempt to make its construction also accessible to non-experts. Next, we consider the subcategory of stratified mixed Tate motives defined for affinely stratified varieties, discuss perverse and parity motives and prove formality results. We combine this with results of Soergel to construct a geometric and graded version of the derived modular category \({\mathcal {O}}(G)\), consisting of rational representations of a semisimple algebraic group \(G/\mathbb {k}\).

Keywords

Motives Representation theory Positive characteristic Reductive groups 

Mathematics Subject Classification

Primary 14F05 Secondary 14C15 14M15 19D45 20G40 

Notes

Acknowledgements

We thank the referee for their very detailed and constructive suggestions improving the exposition of this article substantially. We would like to thank Wolfgang Soergel for many encouraging and illuminating discussions. The first author thanks Markus Spitzweck and Matthias Wendt for instructive email exchanges about motivic six functors. We also thank Oliver Braunling, Brad Drew, Thomas Geisser, Lars Thorge Jensen and Simon Pepin Lehalleur for their valuable input. The second author thanks the first author for the opportunity to work on this project, and for many stimulating questions and discussions. The first author was financially supported by the DFG Graduiertenkolleg 1821 “Cohomological Methods in Geometry”.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California Los AngelesLos AngelesUSA
  2. 2.Department of MathematicsTokyo Institute of TechnologyTokyoJapan

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