Newforms of Saito-Kurokawa lifts

Abstract

A new-and old-form theory for Bessel periods of (cuspidal automorphic) Saito-Kurokawa representations π = ⊗vπv of PGSp 4 is given. We introduce arithmetic subgroups so that a local Bessel vector fixed by the subgroup indexed by the conductor of πv is unique up to scalars. This vector is called the local newform of πv. The global Langlands L-function of a holomorphic Saito-Kurokawa representation coincides with a canonically settled Piatetski-Shapiro zeta integral of the global newform.