  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.
