A Hecke action on $G_1T$-modules

We give an action of the Hecke category on the principal block $\mathrm{Rep}_0(G_1T)$ of $G_1T$-modules where $G$ is a connected reductive group over an algebraically closed field of characteristic $p > 0$, $T$ a maximal torus of $G$ and $G_1$ the Frobenius kernel of $G$. To define it, we define a new category with a Hecke action which is equivalent to the combinatorial category defined by Andersen-Jantzen-Soergel.