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

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

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