Certain Fourier operators on GL1 and local Langlands gamma functions

Abstract

. 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 [JL21], 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 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 \ A . Such general formulae on GL 1 , as a vast generalization of the classical Poisson summation formula, are expected to be responsible for the Langlands conjecture ([ 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 [JL21] 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 σ ν ,ρ,ψ ν ( x ) on GL 1 that represent the ( σ ν , ρ )-Fourier operators 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 functions. As a consequence, we show that any local Langlands γ -functions are the gamma functions in the sense of I. and I. Piatetski-Shapiro in [GGPS] and of A. Weil in [W66].