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Abstract
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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
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
.
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Keywords
Langlands–Shahidi method,
Jacquet–Piatetski-Shapiro–Shalika, Rankin–Selberg, Gamma
factors, Bessel function, Kloosterman sums, Representations
of GLn(Fq)
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Mathematical Subject Classification
Primary: 11F66, 11T24, 20C33
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Milestones
Received: 16 January 2023
Revised: 1 September 2023
Accepted: 18 September 2023
Published: 31 December 2023
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© 2023 MSP (Mathematical Sciences
Publishers). |
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