CERTAIN FOURIER OPERATORS AND THEIR ASSOCIATED POISSON SUMMATION FORMULAE ON GL₁

We explore the possibility of using harmonic analysis on GL₁ to understand Langlands automorphic L-functions in general, as a vast generalization of the PhD Thesis of J. Tate in 1950. 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 S_σ,ρ(A^×) of (σ, ρ)-Schwartz functions on A^× and (σ, ρ)-Fourier operator F_σ,ρ,ψ that takes S_σ,ρ(A^×) to S_{σ,ρ e} (A^×), where σe is the contragredient of σ. By assuming the local Langlands functoriality for the pair (G, ρ), we show that the (σ, ρ)-theta functions 2_σ,ρ(x, φ): = ^P _α∈k× φ(αx) converge absolutely for all φ ∈ S_σ,ρ(A^×). We state conjectures on the (σ, ρ)-Poisson summation formula on GL₁, and prove them in the case where G = GL_n and ρ is the standard representation of GL_n(C). This is done with the help of results of Godement and Jacquet (1972). As an application, we provide a spectral interpretation of the critical zeros of the standard L-functions L(s, π × χ) for any irreducible cuspidal automorphic representation π of GL_n(A) and idele class character χ of k, extending theorems of C. Soulé (2001) and A. Connes (1999). Other applications are in the introduction.