## Abstract

We explore the possibility of using harmonic analysis on GL_{1} 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 GL_{1}, 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.

Original language | English (US) |
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Pages (from-to) | 301-372 |

Number of pages | 72 |

Journal | Pacific Journal of Mathematics |

Volume | 326 |

Issue number | 2 |

DOIs | |

State | Published - 2024 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© (2024), (Mathematical Sciences Publishers). All Rights Reserved.

## Keywords

- automorphic
- Fourier operator
- invariant distribution
- Poisson summation formula

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