Let G be a simply connected semisimple algebraic group over a field of characteristic greater than the Coxeter number. We construct a monoidal action of the diagrammatic Hecke category on the principal block Rep 0 pGq of ReppGq by wall-crossing functors. This action was conjectured to exist by Riche-Williamson [RW15]. Our method uses constructible sheaves and relies on Smith-Treumann theory as applied in [RW20].

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

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