On top Fourier coefficients of certain automorphic representations of GLn

Abstract

In this paper, we study top Fourier coefficients of certain automorphic representations of ${\mathrm{GL}}_n(\mathbb{A})$. In particular, we prove a conjecture of Jiang on top Fourier coefficients of isobaric automorphic representations of ${\mathrm{GL}}_n(\mathbb{A})$ of form $$\Delta ({\tau }_1,b_1)⊞\Delta ({\tau }_2,b_2)⊞\cdots ⊞\Delta ({\tau }_r,b_r),$$ where $\Delta ({\tau }_i,b_i)$'s are Speh representations in the discrete spectrum of ${\mathrm{GL}}_{a_i{b}_i}(\mathbb{A})$ with ${\tau }_i$'s being unitary cuspidal representations of ${\mathrm{GL}}_{a_i}(\mathbb{A})$, and $n={\sum }_{i=1}^r{a}_i{b}_i$. Endoscopic lifting images of the 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.

Comments are welcome