A coherent categorification of the based ring of the lowest two-sided cell

We give a partial coherent categorification of $J_0$, the based ring of the lowest two sided cell of an affine Weyl group, equipped with a monoidal functor from the category of coherent sheaves on the derived Steinberg variety. We show that our categorification acts on natural coherent categorifications of the Iwahori invariants of the Schwartz space of the basic affine space. In low rank cases, we construct complexes that lift the basis elements $t_w$ of $J_0$ and their structure constants.