Hecke action on the principal block

Abstract

<jats:p>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 <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0010437X22007436_inline1.png" /><jats:tex-math>$p$</jats:tex-math></jats:alternatives></jats:inline-formula>-Kazhdan–Lusztig polynomials.</jats:p>