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.
Categories: math.NT math.RT