An Analogue of Bernstein-Zelevinsky Derivatives to Automorphic Forms

In this paper, a construction to imitate the Bernstein-Zelevinsky derivative for automorphic representations on $GL_n(\mathbb{A})$ is introduced. We will later consider the induced representation $$I(\tau_1,\tau_2;\underline{s}) = \mathrm{Ind}_{P_{[n_1,n_2]}}^{G_n}(\Delta(\tau_1,n_1)|\cdot|^{s_1}\boxtimes \Delta(\tau_2,n_2)|\cdot|^{s_2}).$$ from the discrete spectrum representations of $GL_n(\mathbb{A})$, and apply our method to study the degenerate Whittaker coefficients of the Eisenstein series constructed from such a representation as well as of its residues. This method can be used to reprove the results on the Whittaker support of automorphic forms of such kind proven by D. Ginzburg, Y. Cai and B. Liu. This method will also yield new results on the Eulerianity of certain degenerate Whittaker coefficients.