© 2018 Mathematical Sciences Publishers. Braverman and Kazhdan have introduced an influential conjecture on local functional equations for general Langlands L-functions. It is related to L. Lafforgue's equally influential conjectural construction of kernels for functorial transfers. We formulate and prove a version of Braverman and Kazhdan's conjecture for spherical representations over an archimedean field that is suitable for application to the trace formula. We then give a global application related to Langlands' beyond endoscopy proposal. It is motivated by Ngô's suggestion that one combine nonabelian Fourier transforms with the trace formula in order to prove the functional equations of Langlands L-functions in general.