On top Fourier coefficients of certain automorphic representations of GLn

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 $$ Delta(tau_1, b_1) boxplus Delta(tau_2, b_2) boxplus cdots boxplus Delta(tau_r, b_r),, $$ where $Delta(tau_i,b_i)$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 $n = sum_{i=1}^r a_ib_i$. 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.