Trace Paley-Wiener theorem for Braverman-Kazhdan's asymptotic Hecke algebra

Let $\mathbf G$ be a reductive algebraic group over a non-archimedean local field $F$ of characteristic zero and let $G=\mathbf G(F)$ be the group of $F$-rational points. Let $\mathcal H(G)$ be the Hecke algebra and let $\mathcal J(G)$ be the asymptotic Hecke algebra, as defined by Braverman and Kazhdan. We classify irreducible representations of $\mathcal J(G)$. As a consequence, we prove a conjecture of Bezrukavnikov-Braverman-Kazhdan that the inclusion $\mathcal H(G)\subset\mathcal J(G)$ induces an isomorphism $\mathcal H(G)/[\mathcal H(G),\mathcal H(G)]\simeq\mathcal J(G)/[\mathcal J(G),\mathcal J(G)]$ on the cocenters. We also provide an explicit description of $\mathcal J(G)$ and the cocenter $\mathcal H(G)/[\mathcal H(G),\mathcal H(G)]$ when $\mathbf G=\mathrm{GL}_n$.