On the conjectures of Braverman-Kazhdan

In this article we prove a conjecture of Braverman and Kazhdan in \cite{BK1} on acyclicity of $\rho$-Bessel sheaves on reductive groups in both $\ell$-adic and de Rham settings. We do so by establishing a vanishing conjecture proposed in \cite{C1}. As a corollary, we obtain a geometric construction of the non-linear Fourier kernels for finite reductive groups as conjectured by Braverman and Kazhdan. The proof of the vanishing conjecture relies on the techniques developed in \cite{BFO} on Drinfeld center of Harish-Chandra bimodules and character D-modules, and a construction of a class of character sheaves in mixed-characteristic.