Abstract: Character sheaves provide a geometric framework for studying characters of finite groups of Lie type. Introduced by Lusztig in the 1980s, they played a central role in the classification of irreducible complex representations of these groups and in the computation of their characters. Although the definition of character sheaves can be formulated with modular coefficients, much of Lusztig's characteristic-zero machinery breaks down in this setting.

In this talk, I will give a streamlined overview of the theory of character sheaves using modern language from categorical representation theory. From this perspective, categorical traces allow one to study character sheaves through monodromic Hecke categories. I will describe two models for these categories: a geometric categorification in terms of parity sheaves and an algebraic categorification in terms of Soergel bimodules. A key advantage of these models is that they are tightly controlled by p-Kazhdan–Lusztig combinatorics. This paves the way for a more algebraic study of modular character sheaves.