# Goals

Understand Bezrukavnikov's "two equivalences".

> [Stefan:] it’s been known since the 80s (actually properly more like the 90s I think) that the affine Hecke algebra and equivariant K-theory of the Steinberg variety of the dual group are isomorphic as Z[q,q^-1]-modules. And people wanted to categorify this. It turns an infinite-dimensional, analytic, problem on a p-adic group (modules over the Hecke algebra) into finite-dimensional complex geometry. Bezrukavnikov’s categorification(s) had to change the coherent side a bit: you need derived schemes instead of schemes for it to work, but it provides a very nice way to translate things. Much is known on the coherent side as far as I can tell. Also his equivalences are compatible with Geometric Satake