  In this paper we construct an action of the affine Hecke category (in its
"Soergel bimodules" incarnation) on the principal block of representations of a
simply-connected semisimple algebraic group over an algebraically closed field
of characteristic bigger than the Coxeter number. This confirms a conjecture of
G. Williamson and the second author, and provides a new proof of the tilting
character formula in terms of antispherical $p$-Kazhdan-Lusztig polynomials.
