On Fourier coefficients of certain residual representations of symplectic groups

TL;DR

This paper studies the Fourier coefficients of residual automorphic representations of symplectic groups, linking their structure to global Arthur parameters and advancing the theory of automorphic descents.

Contribution

It provides new insights into the structure of Fourier coefficients of residual representations related to symplectic groups, partially confirming a conjecture and paving the way for more general automorphic descent methods.

Findings

Partial confirmation of a conjecture relating Arthur parameters and Fourier coefficients.

Identification of Fourier coefficient structures for certain residual representations.

Foundation for future automorphic descent techniques for symplectic groups.

Abstract

In the theory of automorphic descents developed by Ginzburg, Rallis and Soudry in [GRS11], the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays important roles. Started from [JLZ13], the authors are looking for more general residual representations, which may yield more general theory of automorphic descents. In this paper, we investigate the structure of Fourier coefficients of certain residual representations of symplectic groups, corresponding to certain interesting families of global Arthur parameters. On one hand, the results partially confirm a conjecture proposed by the first named author in [J14] on relations between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the corresponding global Arthur packets. On the other hand, the results of this paper can be regarded as a first step towards more general automorphic descents for symplectic groups, which will be considered in our future work.