The Hecke category action on the principal block via Smith theory

Wall-crossing functors on the principal block of category OO give an action of the (finite) Hecke category. If one knows enough about the Hecke category, one can deduce the Kazhdan-Lusztig conjectures from the existence of this action. This is a simple example of the power of categorification. In 2013, Riche and I conjectured that something similar is true for the principal block of reductive algebraic groups: namely that wall-crossing functors give an action of the (affine) Hecke category. We showed that the conjecture implies several rather deep statements in representation theory (mod pp analogues of the Kazhdan-Lusztig conjectures). Recently, this conjecture has been proved in two different ways: the first (by Bezrukavnikov and Riche) via mod pp localization and the second (by my student Josh Ciappara) via Smith theory. I will give an outline of Josh's proof. Bezrukavnikov and Riche's proof will be discussed later in the seminar.