Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL_2

We introduce a local zeta-function for an irreducible admissible supercuspidal representation \(\pi\) of the metaplectic double cover of \(\SL_2\) over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is gi...

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Bibliographic Details
Published inarXiv.org
Main Authors Oshita, Kazuki, Tsuzuki, Masao
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 25.05.2023
Subjects
Bessel functions
Functional equations
Mathematical analysis
Matrices (mathematics)
Vector spaces
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Summary:We introduce a local zeta-function for an irreducible admissible supercuspidal representation \(\pi\) of the metaplectic double cover of \(\SL_2\) over a non-archimedean local field of characteristic zero. We prove a functional equation of the local zeta-functions showing that the gamma factor is given by a Mellin type transform of the Bessel function of \(\pi\). We obtain an expression of the gamma factor, which shows its entireness on \(\C\). Moreover, we show that, through the local theta-correspondence, the local zeta-function on the covering group is essentially identified with the local zeta-integral for spherical functions on \({\rm PGL}_2\cong {\rm SO}_3\) associated with the prehomogenous vector space of binary symmetric matrices.
ISSN:2331-8422