Certain Fourier Operators on $mathrm{GL}_1$ and Local Langlands Gamma functions

For a split reductive group $G$ over a number field $k$, let $rho$ be an $n$-dimensional complex representation of its complex dual group $G^vee(mathbb{C})$. For any irreducible cuspidal automorphic representation $sigma$ of $G(mathbb{A})$, where $mathbb{A}$ is the ring of adeles of $k$, in cite{JL21}, the authors introduce the $(sigma,rho)$-Schwartz space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho}$, and study the $(sigma,rho,psi)$-Poisson summation formula on $mathrm{GL}_1$, under the assumption that the local Langlands functoriality holds for the pair $(G,rho)$ at all local places of $k$, where $psi$ is a non-trivial additive character of $kbackslashmathbb{A}$. Such general formulae on $mathrm{GL}_1$, as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture (cite{L70}) on global functional equation for the automorphic $L$-functions $L(s,sigma,rho)$. In order to understand such Poisson summation formulae, we continue with cite{JL21} and develop a further local theory related to the $(sigma,rho)$-Schwartz space $mathcal{S}_{sigma,rho}(mathbb{A}^times)$ and $(sigma,rho)$-Fourier operator $mathcal{F}_{sigma,rho}$. More precisely, over any local field $k_ u$ of $k$, we define distribution kernel functions $k_{sigma_ u,rho,psi_ u }(x)$ on $mathrm{GL}_1$ that represent the $(sigma_ u,rho)$-Fourier operators $mathcal{F}_{sigma_ u,rho,psi_ u}$ as convolution integral operators, i.e. generalized Hankel transforms, and the local Langlands $gamma$-functions $gamma(s,sigma_ u,rho,psi_ u)$ as Mellin transform of the kernel function. As consequence, we show that any local Langlands $gamma$-functions are the gamma functions in the sense of Gelfand, Graev, and Piatetski-Shapiro in cite{GGPS}.