Functional equations and gamma factors of local zeta functions for the metaplectic cover of SL₂

2023/05/25 by Kazuki Oshita, Oshita, Kazuki, Masao Tsuzuki +1
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2305.16537

openalex publication_date 2023/05/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a local zeta-function for an irreducible admissible supercuspidal representation π of the metaplectic double cover of \SL₂ 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 π. 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 PGL₂≅ \rm SO₃ associated with the prehomogenous vector space of binary symmetric matrices.

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