Local and global Maass relations (expanded version)

TL;DR

This paper characterizes certain representations of GSp(4,F) using local relations analogous to classical Maass relations, providing a new representation-theoretic perspective on Saito-Kurokawa lifts.

Contribution

It introduces local relations for GSp(4,F) representations that generalize Maass relations, offering a new approach to understanding Saito-Kurokawa lifts without relying on Fourier coefficient constructions.

Findings

Derived local relations analogous to Maass relations for GSp(4,F) representations.

Connected local relations to classical Maass relations in a representation-theoretic framework.

Provided a new characterization of Saito-Kurokawa lifts using Fourier coefficient averages.

Abstract

We characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their spherical vector in a special Bessel model. These local relations are analogous to the Maass relations satisfied by the Fourier coefficients of Siegel modular forms of degree 2 in the image of the Saito-Kurokawa lifting. We show how the classical Maass relations can be deduced from the local relations in a representation theoretic way, without recourse to the construction of Saito-Kurokawa lifts in terms of Fourier coefficients of half-integral weight modular forms or Jacobi forms. As an additional application of our methods, we give a new characterization of Saito-Kurokawa lifts involving a certain average of Fourier coefficients.