Hecke action on the principal block

Abstract

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.

v1: 57 pages; v2: definition of completed algebras (and categories of modules) corrected, and some minor technical assumptions removed, 57 pages; v3: revision after a referee report: various clarifications and addition of an index of notation, 68 pages; v4: correction of typos, final version to appear in Compositio Math, 68 pages