CERTAIN FOURIER OPERATORS ON GL₁ 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^∨(C). For any irreducible cuspidal automorphic representation σ of G(A), where A is the ring of adeles of k, in [Jiang and Luo 2021], the authors introduce the (σ, ρ)-Schwartz space S_σ,ρ(A^×) and (σ, ρ)-Fourier operator F_σ,ρ, and study the (σ, ρ,ψ)- Poisson summation formula on GL1, under the assumption that the local Langlands functoriality holds for the pair (G, ρ) at all local places of k, where ψ is a nontrivial additive character of k\A. Such general formulas on GL1, as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture [Langlands 1970] on global functional equation for the automorphic L-functions L(s, σ, ρ). In order to understand such Poisson summation formulas, we continue with Jiang and Luo [2021] and develop a further local theory related to the (σ, ρ)-Schwartz space S_σ,ρ(A^×) and (σ, ρ)-Fourier operator F_σ,ρ. More precisely, over any local field k_ν of k, we define distribution kernel functions k_{σν,ρ,ψν (x)} on GL₁ that represent the (σ_ν, ρ)-Fourier operators F_σν,ρ,ψν 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 functions. As a consequence, we show that any local Langlands γ - functions are the gamma functions in the sense of I. Gelfand, M. Graev, and I. Piatetski-Shapiro [Gelfand et al. 2016] and of A.