On gamma factors for representations of finite general linear groups

Abstract

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

30 pages. Comments are welcome. v2: This version contains all changes that were made for submission in Essential Number Theory. Numbering has changed from v1

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• Local Theta correspondence of Tempered Representations and Langlands parameters
• Towards the Jacquet Conjecture on the Local Converse Problem for $p$-adic ${\mathrm{GL}}_n$
• Exterior square gamma factors for cuspidal representations of ${\mathrm{GL}}_n$: finite field analogs and level zero representations
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• On values of the Bessel function for generic representations of finite general linear groups

Cited by in corpus (1)

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