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 recently 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). I will also talk about applications in
Springer Theory.