The notion of asymptotic Hecke algebra originates to the works of Josef and Lusztig. However, no geometric description of this object has been found up to the very recent time.
In my talk, I will proof such a description (conjectured by Qiu and Xi), and explain some applications. In particular, the description of the cocenter of usual affine Hecke algebra will naturally appear.
This is a joint work with Roman Bezrukavnikov and Vasily Krylov.