The Hecke category is a monoidal category which controls several categories in representation theory. For example, appearances of Kazhdan-Lusztig polynomials usually point to an action of the Hecke category. I will discuss progress (partly joint with Ben Elias and Xuhua He) and conjectures (joint with Simon Riche) in the representation of reductive algebraic groups in characteristic p, which arise from studying the Hecke category modulo p.

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