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On gamma factors for representations of finite general linear groups

David Soudry and Elad Zelingher

Vol. 2 (2023), No. 1, 45–82
Abstract

We use the Langlands–Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of GL n(𝔽q) and GL m(𝔽q). We prove that the Shahidi gamma factor is multiplicative and show that it is related to the Jacquet–Piatetski-Shapiro–Shalika gamma factor. As an application, we prove a converse theorem based on the absolute value of the Shahidi gamma factor, and improve the converse theorem of Nien. As another application, we give explicit formulas for special values of the Bessel function of an irreducible generic representation of GL n(𝔽q).

Keywords
Langlands–Shahidi method, Jacquet–Piatetski-Shapiro–Shalika, Rankin–Selberg, Gamma factors, Bessel function, Kloosterman sums, Representations of GLn(Fq)
Mathematical Subject Classification
Primary: 11F66, 11T24, 20C33
Milestones
Received: 16 January 2023
Revised: 1 September 2023
Accepted: 18 September 2023
Published: 31 December 2023
Authors
David Soudry
School of Mathematical Sciences
Sackler Faculty of Exact Sciences
Tel Aviv University
Tel Aviv
Israel
Elad Zelingher
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States