On the existence of a p-adic metaplectic Tate-type $\widetilde {\gamma}$-factor

Abstract

Let $\mathbb{F}$ be a p-adic field, let χ be a character of $\mathbb{F}^{*}$, let ψ be a character of $\mathbb{F}$ and let $\gamma_{\psi}^{-1}$ be the normalized Weil factor associated with a character of second degree. We prove here that one can define a meromorphic function $\widetilde{\gamma}(\chi ,s,\psi)$ via a similar functional equation to the one used for the definition of the Tate γ-factor replacing the role of the Fourier transform with an integration against $\psi\cdot\gamma_{\psi}^{-1}$. It turns out that γ and $\widetilde{\gamma}$ have similar integral representations. Furthermore, $\widetilde{\gamma}$ has a relation to Shahidi‘s metaplectic local coefficient which is similar to the relation γ has with (the non-metalpectic) Shahidi‘s local coefficient. Up to an exponential factor, $\widetilde{\gamma}(\chi,s,\psi)$ is equal to the ratio $\frac{\gamma(\chi^{2},2s,\psi)}{\gamma(\chi,s+\frac{1}{2},\psi)}$.