Bezrukavnikov's equivalence over the integers

In joint work with Gurbir Dhillon, we prove that the universal monodromic affine Hecke category is equivalent to ind-coherent sheaves on the Steinberg stack, an extension of Bezrukavnikov's equivalence. I will describe the proof, assuming as input a coherent description of the Iwahori-Whittaker category. The main ingredients are categorical representation theory and analysis of t-structures. Working over the integers presents some new complications because then perfect complexes on a stack are typically not compact.