Mixed Motives and Geometric Representation Theory in Equal Characteristic

Let k be a field of characteristic p. We introduce a formalism of mixed sheaves with coefficients in k and apply it in representation theory. We construct a system of k-linear triangulated category of motives on schemes over Fp, 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 O(G), consisting of rational representations of a semisimple algebraic group G/k.