An Analogue of Bernstein-Zelevinsky Derivatives to Automorphic Forms

TL;DR

This paper introduces a method to mimic Bernstein-Zelevinsky derivatives for automorphic forms on GL_n, enabling new insights into their Whittaker coefficients and support, with applications to Eisenstein series and residues.

Contribution

It develops a novel approach to analyze automorphic representations by imitating Bernstein-Zelevinsky derivatives, extending understanding of their Whittaker coefficients and support.

Findings

Reproves known results on Whittaker support.

Provides new results on Eulerianity of degenerate Whittaker coefficients.

Applies method to Eisenstein series and residues.

Abstract

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.