Tate \(\gamma \)-factor, Weil index, and the metaplectic \(\widetilde{\gamma} \)-factor
The author mainly demonstrated the following equality in this well-written short paper: \[\widetilde{\gamma}(1-s,\chi^{-1}, \psi)=\gamma(\psi) \chi(-1) \frac{\gamma(s+1/2, \chi, \psi)}{\gamma(2s,\chi^2,\psi_2)}.\] In this equality, \(\gamma(\psi)\) is the Weil index, which appears in the study of Weil representations of the metaplectic group \(\widetilde{Sp_{2n}}(F)\). [\textit{A. Weil}, Acta Math. 111, 143--211 (1964; Zbl 0203.03305); \textit{R. Ranga Rao}, Pac. J. Math. 157, No. 2, 335--371 (1993; Zbl 0794.58017); \textit{P. Perrin}, Lect. Notes Math. 880, 370--407 (1981; Zbl 0462.22008)]. \(\gamma(s,\gamma, \psi)\) is the usual Tate \(\gamma\)-factor, which appears in the study of a functional equation in the Langlands program [\textit{J. W. S. Cassels} (ed.) and \textit{A. Fröhlich} (ed.), Algebraic number theory. with the support of the International Mathematical Union. London and New York: Academic Press 1967 (1967; Zbl 0153.07403); \textit{H. Jacquet} and \textit{R. P. Langlands}, Automorphic forms on \(\mathrm{GL} (2)\). Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0236.12010); \textit{R. Godement} and \textit{H. Jacquet}, Zeta functions of simple algebras. Springer, Cham (1972; Zbl 0244.12011); \textit{P. Deligne} and \textit{G. Henniart}, Invent. Math. 64, 89--118 (1981; Zbl 0442.12012); \textit{F. Shahidi}, Duke Math. J. 52, 973--1007 (1985; Zbl 0674.10027); \textit{J. W. Cogdell} et al., Lectures on automorphic \(L\)-functions. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1066.11021)]. \(\widetilde{\gamma}(s,\chi, \psi)\) is called the metaplectic \(\widetilde{\gamma}\)-factor. The exact definition is given in a separate paper by the author [Ramanujan J. 26, No. 1, 45--53 (2011; Zbl 1245.11066)]. For more information on the history of this factor, one can see the introduction. As mentioned in the introduction, the computation of an integral is required to prove the identity. The integral comes from an unpublished note written by \textit{W. J. Sweet jun.} [J. Number Theory 55, No. 2, 222--260 (1995; Zbl 0862.14015)].
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