Fourier Coefficients and Cuspidal Spectrum for Symplectic Groups

Abstract

J. Arthur (The endoscopic classification of representations: orthogonal and Symplectic groups. Colloquium Publication, vol 61. American Mathematical Society, 2013) classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their implication to the structure of the cuspidal spectrum for symplectic groups (Jiang, Automorphic integral transforms for classical groups I: endoscopy correspondences. In: Automorphic forms and related geometry: assessing the legacy of I.I. Piatetski-Shapiro. Contemp. Math., vol 614, pp 179–242. AMS, 2014; Jiang and Liu, Fourier coefficients for automorphic forms on quasisplit classical groups. In: Advances in the theory of automorphic forms and their L-functions. Contemp. Math., vol 664, pp 187–208. AMS, 2016). As a result, we obtain certain characterization and construction of small cuspidal automorphic representations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cuspidal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak (Notes on the generalized Ramanujan conjectures. In: Harmonic analysis, the trace formula, and Shimura varieties. Clay Math. Proc., vol 4, pp 659–685. Amer. Math. Soc., Providence, RI, 2005).