Whittaker-Fourier coefficients of cusp forms on $\widetilde{Sp}_n$: reduction to a local statement

Abstract

In a previous paper we formulated an analogue of the Ichino-Ikeda conjectures for Whittaker-Fourier coefficients of cusp forms on quasi-split groups, as well as the metaplectic group of arbitrary rank. In this paper we reduce the conjecture for the metaplectic group to a local conjectural identity. We motivate this conjecture by giving a heuristic argument for the case $\widetilde{SL}_2$. In a subsequent paper we will prove the local identity in the $p$-adic case.