A Hecke action on $G_{1}T$-modules

We give an action of the Hecke category on the principal block ${\mathrm{R}\mathrm{e}\mathrm{p}}_0(G_{1}T)$ of $G_{1}T$-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.