Newforms of Saito-Kurokawa lifts

A newand old-form theory for Bessel periods of Saito-Kurokawa representations π = ⊗vπv of PGSp4(A) 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. The global Langlands L-function of a holomorphic Saito-Kurokawa representation coincides with a canonically settled Piatetski-Shapiro zeta integral of the newform.

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