Spectral reciprocity and non-vanishing of L-functions

We develop a reciprocity formula for a spectral sum over central values
of L-functions on GL(4) x GL(2). As an application we show that for any
self-dual spherical cuspidal automorphic representation Pi on GL(4),
there exists a self-dual representation pi on GL(2) such that L(1/2, Pi
x pi) does not vanish. An important ingredient is a functional equation
of a certain double Dirichlet series involving Kloosterman sums and
GL(4) Hecke eigenvalues. (Joint work with X. Li and S. Miller)