On Fourier coefficients of certain residual representations of symplectic groups

Dihua Jiang and Baiying Liu

Vol. 281 (2016), No. 2, 421–466

DOI: 10.2140/pjm.2016.281.421

Abstract

In the theory of automorphic descents developed by Ginzburg, Rallis, and Soudry in The descent map from automorphic representations of $GL\left(n\right)$ to classical groups (World Scientific, 2011), the structure of Fourier coefficients of the residual representations of certain special Eisenstein series plays an essential role. In a series of papers starting with Pacific J. Math. 264:1 (2013), 83–123, we have looked for more general residual representations, which may yield a more general theory of automorphic descents. We continue this program here, investigating the structure of Fourier coefficients of certain residual representations of symplectic groups, associated with certain interesting families of global Arthur parameters. The results partially confirm a conjecture proposed by Jiang in Contemp. Math. 614 (2014), 179–242 on relations between the global Arthur parameters and the structure of Fourier coefficients of the automorphic representations in the associated global Arthur packets. The results of this paper can also be regarded as a first step towards more general automorphic descents for symplectic groups, which will be considered in our future work.

Keywords

Arthur parameters, Fourier coefficients, unipotent orbits, automorphic forms

Mathematical Subject Classification 2010

Primary: 11F70, 22E55

Secondary: 11F30

Milestones

Received: 21 October 2014

Revised: 12 June 2015

Accepted: 13 June 2015

Published: 16 February 2016