Trace Paley–Wiener theorem for Braverman–Kazhdan’s Asymptotic Hecke Algebra

The full Hecke algebra of a p-adic group G is complicated, but at Iwahori level, it is the specialization of an affine Hecke algebra at v = q. Lusztig defined the asymptotic Hecke algebra, which is a “limit” of the affine Hecke algebra as v goes to infinity. Although the “limiting process” does not make sense for the full Hecke algebra, Braverman and Kazhdan were able to extend the asymptotic Hecke algebra to arbitrary level. I will explain a proof of a conjecture by Bezrukavnikov, Braverman, and Kazhdan that the cocenter of the Hecke algebra is isomorphic to the cocenter of the asymptotic Hecke algebra.