On Whittaker models and the vanishing of Fourier coefficients of cusp forms (Q1110567)

Examples are given of cuspidal representations of the metaplectic group \(\overline{SL}_ 2\) which have a \(\psi\)-Whittaker model but all whose \(\psi\)-Fourier coefficients vanish. From Waldspurger's results on the theta-correspondence between \(\overline{SL}_ 2\) and \(PGL_ 2\) it follows that such examples will correspond to cuspidal representations \(\pi\) of \(PGL_ 2\) for which \(L(\pi,)=0\), \(\epsilon (\pi,)=1\). For the latter two explicit examples are given.