  In this paper we propose a new approach to tilting modules for reductive
algebraic groups in positive characteristic. We conjecture that translation
functors give an action of the (diagrammatic) Hecke category of the affine Weyl
group on the principal block. Our conjecture implies character formulas for the
simple and tilting modules in terms of the p-canonical basis, as well as a
description of the principal block as the anti-spherical quotient of the Hecke
category. We prove our conjecture for GL_n using the theory of 2-Kac-Moody
actions. Finally, we prove that the diagrammatic Hecke category of a general
crystallographic Coxeter group may be described in terms of parity complexes on
the flag variety of the corresponding Kac-Moody group.
