A Hecke action on the principal block of a semisimple algebraic group

I will explain the construction of an action of the Hecke category on the principal block of representations of a connected reductive algebraic group over an algebraically closed field of positive characteristic, obtained in joint work with Roman Bezrukavnikov. As noticed in earlier work with Geordie Williamson, the existence of this action has nice consequences, in particular a character formula for indecomposable tilting modules in terms of p-Kazhdan-Lusztig polynomials.