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 $k\backslash\mathbb{A}$. 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 $k_\nu$ of $k$, we define distribution kernel functions $k_{\sigma_\nu,\rho,\psi_\nu }(x)$ on $\mathrm{GL}_1$ that represent the $(\sigma_\nu,\rho)$-Fourier operators $\mathcal{F}_{\sigma_\nu,\rho,\psi_\nu}$ as convolution integral operators, i.e. generalized Hankel transforms, and the local Langlands $\gamma$-functions $\gamma(s,\sigma_\nu,\rho,\psi_\nu)$ 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}.