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On gamma factors for representations of finite general linear groups

David Soudry, Elad Zelingher

TL;DR

This work develops a finite-field analogue of Shahidi's local coefficients by defining a Shahidi gamma factor for pairs of irreducible generic representations of GL_n and GL_m over _q via the Langlands--Shahidi method. The gamma factor is proven to be multiplicative and related to the Jacquet--Piatetski-Shapiro--Shalika gamma factor, enabling a new, group-theoretic converse theorem based on the absolute value of the gamma factor. The authors also provide explicit Bessel-function expressions for the gamma factor across cases n>m and n=m, and derive practical formulas for special values of Bessel functions, including two- and three-block scenarios. These results yield quantitative insights into the cuspidal support and extend classical Rankin--Selberg-type results to finite fields, with consequences for classification of representations and explicit evaluation of key functions.

Abstract

We use the Langlands--Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_m\left(\mathbb{F}_q\right)$. 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 $\operatorname{GL}_n\left(\mathbb{F}_q\right)$.

On gamma factors for representations of finite general linear groups

TL;DR

This work develops a finite-field analogue of Shahidi's local coefficients by defining a Shahidi gamma factor for pairs of irreducible generic representations of GL_n and GL_m over _q via the Langlands--Shahidi method. The gamma factor is proven to be multiplicative and related to the Jacquet--Piatetski-Shapiro--Shalika gamma factor, enabling a new, group-theoretic converse theorem based on the absolute value of the gamma factor. The authors also provide explicit Bessel-function expressions for the gamma factor across cases n>m and n=m, and derive practical formulas for special values of Bessel functions, including two- and three-block scenarios. These results yield quantitative insights into the cuspidal support and extend classical Rankin--Selberg-type results to finite fields, with consequences for classification of representations and explicit evaluation of key functions.

Abstract

We use the Langlands--Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of and . 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 .
Paper Structure (28 sections, 39 theorems, 177 equations)

This paper contains 28 sections, 39 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic induction
  4. Generic representations
  5. Whittaker models and Bessel functions
  6. Jacquet--Piatetski-Shapiro--Shalika gamma factors
  7. The case $n > m$
  8. The case $n = m$
  9. Shahidi gamma factors (local coefficients)
  10. The intertwining operator
  11. The Shahidi gamma factor
  12. Dependence on $\psi$
  13. Relation between $\Gamma(\pi \times \sigma, \psi)$ and $\Gamma({\sigma}^{\vee} \times {\pi}^{\vee}, {\psi^{-1}})$
  14. Multiplicativity of Gamma factors
  15. Transitivity of parabolic induction
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\pi$, $\sigma_1$ and $\sigma_2$ be representations of Whittaker type of $\mathrm{GL}_n\left(\mathbb{F}\right)$, $\mathrm{GL}_{m_1}\left(\mathbb{F}\right)$ and $\mathrm{GL}_{m_2}\left(\mathbb{F}\right)$, respectively. Then

Theorems & Definitions (68)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Remark 2.4
  • Theorem 2.5: Nien14
  • ...and 58 more