Abstract
The affine Hecke category is a categorical version of the affine Hecke algebra associated with a reductive group (which controls the admissible representations of the corresponding p-adic group generated by the fixed points of an Iwahori subgroup). This category admits several incarnations (in particular, in terms of coherent bundles on a Steinberg variety, and in terms of constructible bundles on a variety of affine flags). I'll explain some work with R. Bezrukavnikov to show that these incarnations are equivalent, and applications in representation theory of reductive groups.
