Mixed Motives and Geometric Representation Theory in Equal Characteristic

TL;DR

This paper develops a formalism of mixed sheaves over fields of characteristic p, constructing a category of motives that facilitates geometric representation theory and extends concepts like Soergel's category to positive characteristic.

Contribution

It introduces a new category of mixed motives in characteristic p, enabling geometric representation theory techniques and constructing a modular version of Soergel's category with a six functor formalism.

Findings

Constructed a $kk$-linear triangulated category of motives on schemes over $ar{F}_p$.

Established a six functors formalism and computed higher Chow groups within this framework.

Built a geometric graded version of Soergel's modular category $O(G)$ in positive characteristic.

Abstract

Let $\mathbb{k}$ be a field of characteristic $p$. We introduce a formalism of mixed sheaves with coefficients in $\mathbb{k}$ and showcase its use in representation theory. More precisely, we construct for all quasi-projective schemes $X$ over an algebraic closure of $\mathbb{F}_p$ a $\mathbb{k}$-linear triangulated category of motives on $X$. Using work of Ayoub (2007), Cisinski-Deglise (2012) and Geisser-Levine (2000), we show that this system of categories has a six functors formalism and computes higher Chow groups. Indeed, it behaves similarly to other categories of sheaves that one is used to. We attempt to make its construction also accessible to non-experts. We then consider the subcategory of stratified mixed Tate motives defined for affinely stratified varieties $X$, discuss perverse and parity motives and prove formality results. As an example, we combine these results and Soergel (2000) to construct a geometric and graded version of Soergel's modular category $\mathscr O(G)$, consisting of rational representations of a split semisimple group $G/\mathbb{k}$, and thereby equip it with a full six functor formalism (see Riche-Soergel-Williamson (2014) and Achar-Riche (2016) for other approaches). The main idea of using motives in geometric representation theory in this way as well as many results about stratified mixed Tate motives are directly borrowed from Soergel and Wendt, who tell the story in characteristic zero.