A Spectral reciprocity formula and non-vanishing for L-functions on GL(4)×GL(2)

Abstract We introduce a new type of summation formula for central values of GL ( 4 ) × GL ( 2 ) L -functions, when varied over Maas forms. By rewriting such a sum in terms of GL ( 4 ) × GL ( 1 ) L -functions and applying a new “balanced” Voronoi formula, the sum can be shown to be equal to a differently-weighted average of the same quantities. By controlling the support of the spectral weighting functions on both sides, this reciprocity formula gives estimates on spectral sums that were previously obtainable only for lower rank groups. The “balanced” Voronoi formula has Kloosterman sums on both sides, and can be thought of as the functional equation of a certain double Dirichlet series involving Kloosterman sums and GL ( 4 ) Hecke eigenvalues. As an application we show that for any self-dual cusp form Π for SL ( 4 , Z ) , there exist infinitely many Maas forms π for SL ( 2 , Z ) such that L ( 1 / 2 , Π × π ) ≠ 0 .