Certain Fourier Operators and their Associated Poisson Summation Formulae on $\mathrm{GL}_1$

In this paper, we explore possibilities to utilize harmonic analysis on GL1 to understand Langlands automorphic L-functions in general, as a vast generalization of the pioneering work of J. Tate ([T50]). For a split reductive group G over a number field k, let G(C) be its complex dual group and ρ be an n-dimensional complex representation of G(C). For any irreducible cuspidal automorphic representation σ of G(A), where A is the ring of adeles of k, we introduce the space Sσ,ρ(A×) of (σ, ρ)-Schwartz functions on A and (σ, ρ)-Fourier operator Fσ,ρ,ψ that takes Sσ,ρ(A×) to Sσ̃,ρ(A×), where σ̃ is the contragredient of σ. By assuming the local Langlands functoriality for the pair (G, ρ), we show (Theorem 5.8) that the (σ, ρ)-theta functions Θσ,ρ(x, φ) := ∑