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Doubling constructions: Global functoriality for non-generic cuspidal representations

Yuanqing Cai, Solomon Friedberg, Eyal Kaplan

Abstract

We study the generalized doubling method for pairs of representations of $G\times GL_k$ where $G$ is a symplectic group, split special orthogonal group or split general spin group. We analyze the poles of the local integrals, and prove that the global completed $L$-function with a cuspidal representation of $GL_k$ twisted by a highly ramified Hecke character is entire. We obtain a new proof of the weak functorial transfer of cuspidal automorphic representations of $G$ to the natural general linear group, which is independent of the trace formula and its prerequisites, by combining our results with the Converse Theorem.

Doubling constructions: Global functoriality for non-generic cuspidal representations

Abstract

We study the generalized doubling method for pairs of representations of where is a symplectic group, split special orthogonal group or split general spin group. We analyze the poles of the local integrals, and prove that the global completed -function with a cuspidal representation of twisted by a highly ramified Hecke character is entire. We obtain a new proof of the weak functorial transfer of cuspidal automorphic representations of to the natural general linear group, which is independent of the trace formula and its prerequisites, by combining our results with the Converse Theorem.

Paper Structure

This paper contains 15 sections, 33 theorems, 184 equations.

Table of Contents

  1. The groups
  2. The Converse Theorem
  3. Local theory
  4. Notation
  5. Functorial transfer
  6. The integral
  7. gamma, $L$- and epsilon-factors
  8. Coarse transfer
  9. The poles of the $L$-factor
  10. Global theory
  11. Notation
  12. The Eisenstein series
  13. The integral
  14. Constructing a weak functorial transfer
  15. The poles of the GL(c) x GL(k) integrals, by Eyal Kaplan

Key Result

Theorem \oldthetheorem

Any irreducible cuspidal automorphic representation $\pi$ of $G(\mathbb{A})$ has a weak functorial transfer $\Pi$ to $\mathop{\mathrm{GL}}\nolimits_N(\mathbb{A})$.

Theorems & Definitions (72)

  • Theorem \oldthetheorem
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  • Definition \oldthetheorem
  • Example \oldthetheorem
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  • Lemma \oldthetheorem
  • Remark \oldthetheorem
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  • ...and 62 more