  Braverman and Kahzdan 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
Ng\^o'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.
