Newforms of Saito-Kurokawa lifts

TL;DR

This paper develops a new- and old-form theory for Bessel periods of Saito-Kurokawa representations, linking local Bessel vectors, arithmetic subgroups, and global L-functions through zeta integrals.

Contribution

It introduces a novel framework for analyzing Bessel periods and establishes a connection between local vectors and global L-functions for Saito-Kurokawa lifts.

Findings

Unique local Bessel vectors fixed by specific subgroups

Global L-function matches Piatetski-Shapiro zeta integral

Newform theory clarifies structure of Saito-Kurokawa representations

Abstract

A new- and old-form theory for Bessel periods of Saito-Kurokawa representations is given. We introduce arithmetic subgroups so that a local Bessel vector fixed by the subgroup indexed by the conductor of the representation 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.