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On values of the Bessel function for generic representations of finite general linear groups

Elad Zelingher

Abstract

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values of the Bessel function can be realized as the coefficients of $L$-functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.

On values of the Bessel function for generic representations of finite general linear groups

Abstract

We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of . We show that special values of the Bessel function can be realized as the coefficients of -functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.
Paper Structure (21 sections, 41 theorems, 181 equations)

This paper contains 21 sections, 41 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Generic representations and their local constants
  3. Macdonald's parametrization of irreducible representations of $\mathrm{GL}_n\left(\mathbb{F}\right)$
  4. Parametrization of irreducible generic representations of $\mathrm{GL}_n \left(\mathbb{F}\right)$
  5. Whittaker models and their Bessel functions
  6. Rankin--Selberg gamma factors
  7. Shahidi gamma factors
  8. A recursive expression for the Bessel function
  9. Epsilon factors
  10. Étale algebras and exotic Kloosterman sums and sheaves
  11. Étale algebras
  12. Tensor product of finite fields
  13. Katz's exotic Kloosterman sums and sheaves
  14. Formulas for special values of the Bessel function
  15. Generic representations of a given type
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any $a \in \mathbb{F}^{\times}$, let $J_{m}\left(\alpha^{-1}, \psi, a\right)$ be the following exotic Kloosterman sum of Katz. Consider the $L$-function associated with this Kloosterman sum and its normalized version Then

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Proposition 2.2: Gelfand70
  • Proposition 2.3: Nien14
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8: Roditty10 or Nien14
  • ...and 54 more