Vol. 281, No. 2, 2016

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On Fourier coefficients of certain residual representations of symplectic groups

Dihua Jiang and Baiying Liu

Vol. 281 (2016), No. 2, 421–466

In the theory of automorphic descents developed by Ginzburg, Rallis, and Soudry in The descent map from automorphic representations of GL(n) 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.

Arthur parameters, Fourier coefficients, unipotent orbits, automorphic forms
Mathematical Subject Classification 2010
Primary: 11F70, 22E55
Secondary: 11F30
Received: 21 October 2014
Revised: 12 June 2015
Accepted: 13 June 2015
Published: 16 February 2016
Dihua Jiang
University of Minnesota
127 Vincent Hall
206 Church St. SE
Minneapolis, MN 55455
United States
Baiying Liu
School of Mathematics
Institute for Advanced Study
Einstein Drive
Princeton, NJ 08540
United States