Exterior square gamma factors for cuspidal representations of GLn: finite field analogs and level-zero representations

Ye, Rongqing; Zelingher, Elad

doi:10.1007/s11856-020-2084-y

Cited by 5 publications

(3 citation statements)

References 10 publications

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“…2 ) , see for instance [44,Lemma 2.29], but we will not need this. Similarly, we have that the summand in ( 2) is invariant under right multiplication by elements of U m , and hence we have by Theorem 2.12 that…”

Section: 5mentioning

confidence: 99%

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

Zelingher¹

2022

Preprint

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We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of GL n (F q ). 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.

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Section: 5mentioning

confidence: 99%

“…The Bessel function of S. I. Gelfand is closely related to gamma factors of the representations, see [32,26] for its relation to the Rankin-Selberg gamma factors, and [44] for its relation to the exterior square gamma factors. The Bessel function served as a key ingredient in Nien's proof of Jacquet's conjecture [26] for finite fields.…”

Section: Introductionmentioning

confidence: 99%

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

Zelingher¹

2022

Preprint

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“…It allows one to attach "local constants" to irreducible representations of the F-points version of the group in consideration. We mention the works [18,16,25,13,14] as examples. These local constants usually encode properties analogous to their local factors counterparts.…”

Section: Introductionmentioning

confidence: 99%

On gamma factors for representations of finite general linear groups

Soudry¹,

Zelingher²

2023

Preprint

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We use the Langlands-Shahidi method in order to define the Shahidi gamma factor for a pair of irreducible generic representations of GL n (F q ) and GL m (F q ). 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 (F q ).

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

Zelingher

2023

Advances in Mathematics

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Exterior square gamma factors for cuspidal representations of GLn: finite field analogs and level-zero representations

Cited by 5 publications

References 10 publications

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

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

On gamma factors for representations of finite general linear groups

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

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