An Analogue of Bernstein-Zelevinsky Derivatives to Automorphic Forms

2022/07/23 by Zhuohui Zhang, Zhang, Zhuohui
Mathematics · #20G05 #22E55 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2207.11543

openalex publication_date 2022/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, a construction to imitate the Bernstein-Zelevinsky derivative for automorphic representations on GL_n(\mathbbA) is introduced. We will later consider the induced representation I(τ₁,τ₂;\underlines) = Ind_P_[n1,n2]^G_n(Δ(τ₁,n₁)|⋅|^s₁\boxtimes Δ(τ₂,n₂)|⋅|^s₂). from the discrete spectrum representations of GL_n(\mathbbA), 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.

Related

• Construction of simple quotients of Bernstein-Zelevinsky derivatives and highest derivative multisegments I: reduction to combinatorics
• Bernstein-Zelevinsky derivatives: a Hecke algebra approach
• Fourier coefficients attached to small automorphic representations of SL_n(\mathbbA)
• Explicit GL(2) trace formulas and uniform, mixed Weyl laws
• Small automorphic representations and degenerate Whittaker vectors