Certain Fourier operators on GL1 and local Langlands gamma functions

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