  We give a 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.
