Many questions about the representation theory of a complex semisimple Lie group can be understood in terms of the category $\mathcal{O}(\mathfrak{g})$ associated to its Lie algebra.

In analogy, Soergel constructed a modular category $\mathcal{O}(G)$ of representations of a reductive algebraic group $G$ over a field in characteristic $p$, which was used by Williamson to construct counterexamples to Lusztig's conjecture ("Williamson's Torsion Explosion").

Both categories are intimately related to the mixed geometry of the flag variety. In characteristic $0$, categories of certain mixed $\ell$-adic sheaves, mixed Hodge modules or stratified mixed Tate motives provide geometric versions of the derived graded category $\mathcal{O}(\mathfrak{g})$ (Beilinson, Ginzburg, Soergel and Wendt).

Using the work of Soergel, we prove analogous statements in characteristic $p$. First, we construct an appropriate formalism of "mixed modular sheaves", using motives in equal characteristic. We then apply this formalism to construct a geometric version of the of the derived graded modular category $\mathcal{O}(G)$. (This is joint work with Shane Kelly).

We will also discuss applications to Springer Theory.