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 $\mathcal{H}(G)\mathrm{/}[\mathcal{H}(G),\mathcal{H}(G)]\simeq \mathcal{J}(G)\mathrm{/}[\mathcal{J}(G),\mathcal{J}(G)]$ on the cocenters. We also provide an explicit description of $\mathcal{J}(G)$ and the cocenter $\mathcal{H}(G)\mathrm{/}[\mathcal{H}(G),\mathcal{H}(G)]$ when $\mathbf{G}={\mathrm{G}\mathrm{L}}_n$.