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

Abstract

We study the top Fourier coefficients of isobaric automorphic representations of $${\mathrm {GL}}_n({\mathbb {A}})$$ of the form $$\begin{aligned} \Pi _{\underline{s}} = {\mathrm {Ind}}^{{\mathrm {GL}}_n({\mathbb {A}})}_{P({\mathbb {A}})} \Delta (\tau _1,b_1) |\cdot |^{s_1} \otimes \cdots \otimes \Delta (\tau _r,b_r) |\cdot |^{s_r}, \end{aligned}$$ where $$s_i\in {\mathbb {C}}$$ , $$\Delta (\tau _i,b_i)$$ ’s are Speh representations in the discrete spectrum of $${\mathrm {GL}}_{a_ib_i}({\mathbb {A}})$$ with $$\tau _i$$ ’s being unitary cuspidal representations of $${\mathrm {GL}}_{a_i}({\mathbb {A}})$$ , and $$n = \sum _{i=1}^r a_ib_i$$ . In particular, we prove a part of a conjecture of Ginzburg, and also a conjecture of Jiang under certain assumptions. The result of this paper will facilitate the study of automorphic forms of classical groups occurring in the discrete spectrum.