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On Ginzburg-Kaplan gamma factors and Bessel-Speh functions for finite general linear groups

Oded Carmon, Elad Zelingher

TL;DR

The article develops a finite-field analogue of tensor product gamma factors for GL_c(\mathbb{F}_q) and GL_k(\mathbb{F}_q) using Ginzburg–Kaplan methods and Speh representations, introducing the GK gamma factor $\Gamma^{\mathrm{GK}}(\pi\times\tau,\psi)$ via Bessel–Speh models. It proves multiplicativity in both arguments and establishes a functional equation, connecting these gamma factors to non-abelian Gauss sums and to the special values of Bessel–Speh functions. A key insight is the link between $\mathcal{K}_{\tau,\psi}$-values and twisted matrix Kloosterman sums, from which the authors derive a multiplicativity identity for matrix Kloosterman sums and present a new converse theorem for generic GL representations over finite fields. The results illuminate the finite-field analogue of doubling-type constructions, with potential implications for level-zero correspondences and explicit harmonic analysis on finite groups of Lie type.

Abstract

We give a new construction of tensor product gamma factors for a pair of irreducible representations of $\operatorname{GL}_c\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_k\left(\mathbb{F}_q\right)$. This construction is a finite field analog of a construction of doubling type due to Kaplan in the local field case and due to Ginzburg in the global case, and it only assumes that one of the representations in question is generic. We use this construction to establish a relation between special values of Bessel functions attached to Speh representations of generic principal series representations and twisted matrix Kloosterman sums. Using this relation, we establish the multiplicativity identity of twisted matrix Kloosterman sums.

On Ginzburg-Kaplan gamma factors and Bessel-Speh functions for finite general linear groups

TL;DR

The article develops a finite-field analogue of tensor product gamma factors for GL_c(\mathbb{F}_q) and GL_k(\mathbb{F}_q) using Ginzburg–Kaplan methods and Speh representations, introducing the GK gamma factor via Bessel–Speh models. It proves multiplicativity in both arguments and establishes a functional equation, connecting these gamma factors to non-abelian Gauss sums and to the special values of Bessel–Speh functions. A key insight is the link between -values and twisted matrix Kloosterman sums, from which the authors derive a multiplicativity identity for matrix Kloosterman sums and present a new converse theorem for generic GL representations over finite fields. The results illuminate the finite-field analogue of doubling-type constructions, with potential implications for level-zero correspondences and explicit harmonic analysis on finite groups of Lie type.

Abstract

We give a new construction of tensor product gamma factors for a pair of irreducible representations of and . This construction is a finite field analog of a construction of doubling type due to Kaplan in the local field case and due to Ginzburg in the global case, and it only assumes that one of the representations in question is generic. We use this construction to establish a relation between special values of Bessel functions attached to Speh representations of generic principal series representations and twisted matrix Kloosterman sums. Using this relation, we establish the multiplicativity identity of twisted matrix Kloosterman sums.
Paper Structure (45 sections, 31 theorems, 276 equations, 1 figure)

This paper contains 45 sections, 31 theorems, 276 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Classical constructions of gamma factors
  3. First family: construction via matrix coefficients
  4. Second family: constructions via Whittaker models
  5. Constructions of doubling type
  6. Main results
  7. Ginzburg--Kaplan gamma factors
  8. Relation to matrix Kloosterman sums
  9. A new converse theorem
  10. Representations of (k,c) type and Speh representations
  11. Parameterization of irreducible representations of general linear groups
  12. Parabolic induction
  13. Cuspidal representations and cuspidal support
  14. Parameters of irreducible representations
  15. Representations of (k,c) type
  16. ...and 30 more sections

Key Result

Theorem 1.1

Suppose that $k \ge 2$. For $W \in \mathcal{W}\left(\Delta\left(\tau, c\right), \psi_{\qty(c^{k})}\right)$ and $f \colon \operatorname{Mat}_c\left(\mathbb{F}\right) \to \mathbb{C}$ let and Suppose that $\pi$ and $\tau^{\vee}$ do not have common elements in their cuspidal supports. Then

Theorems & Definitions (56)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • proof
  • ...and 46 more