On top Fourier coefficients of certain automorphic representations of $${\mathrm {GL}}_n$$

Abstract

In this paper, we study top Fourier coefficients of certain automorphic representations of GL n (A). In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of GL n (A) of formwhere ∆(τ i , b i )'s are Speh representations in the discrete spectrum of GL aibi (A) with τ i 's being unitary cuspidal representations of GL ai (A), and n = r i=1 a i b i . Endoscopic lifting images of discrete spectrum of classical groups form a special class of such representations. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.

An Analogue of Bernstein-Zelevinsky Derivatives to Automorphic Forms

“…• [Cai18] for the orbit of the top Fourier coefficient of degenerate Eisenstein series of GL(n), and in [LX20] for the orbit corresponding to the top Fourier coefficient of isobaric sum representations. In this section, we will provide a global, inductive proof to the vanishing of the Fourier coefficients associated with larger orbits, as well as an argument for the Eulerianity of certain Fourier-Whittaker coefficients.…”

“…This method can be applied to prove results regarding nilpotent orbit associated to the top Fourier coefficient. Some of them have already been provided in [Gin06], [Liu13], [Cai18] and [LX20].…”

Some Combinatorics in the Cancellation of Poles of Eisenstein Series for $GL(n,\mathbb{A}_\mathbb{Q})$

“…⊗ ∆(τ r , n r )| • | sr ) . For τ i mutually distinct and tempered at all local places, assuming L( 1 2 , τ i × τj ) = 0 and s lying in the domain in which the Eisenstein series E(•, s) : I(τ , s) → A GN has no poles, [LX20] showed that the Whittaker support of I(τ , s) corresponds to the partition [a n1 1 ] + . .…”

On Wave Front Sets of Global Arthur Packets of Classical Groups: Upper Bound

“…This fundamental result has been indispensable in the theory, especially the theory of automorphic L-functions. The theorem of Piatetski-Shapiro and Shalika has been extended to the discrete spectrum of GL n (A) in [JL13] and to the isobaric sum automorphic spectrum of GL n (A) in [LX21].…”