Braverman and Kazhdan have introduced an influential
conjecture on local functional equations for general Langlands
-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
-functions
in general.
