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Fourier-Jacobi models for real symplectic-metaplectic groups: the basic case

Cheng Chen, Rui Chen, Jialiang Zou

TL;DR

This work settles the basic tempered case of the local Gan-Gross-Prasad conjecture for Fourier-Jacobi models on real symplectic-metaplectic groups by combining the real-LLC framework with stable-range theta lifts and seesaw techniques. The authors prove multiplicity one and an epsilon-dichotomy across tempered L-packets, and establish the existence of Fourier-Jacobi models within each packet, leveraging the Shimura–Waldspurger correspondence and Jacobi–Weil representations. The approach navigates non-tempered reductions through stable-range arguments and a peeling-off strategy, building a bridge between tempered Fourier–Jacobi and Bessel models. Together with prior non-Archimedean and complex results, this work completes the archimedean instance of the conjecture and sharpens the LLC/Prasad-parameter dictionary for these dual pairs.

Abstract

In this paper, we generalize the method of Gan-Ichino and Atobe in [GI16][A18] to the field of real numbers and prove the basic tempered case of the local Gan-Gross-Prasad conjecture for Fourier-Jacobi models of symplectic-metaplectic groups, based on the tempered case of the conjecture for Bessel models proved in [CL22] by Chen-Luo.

Fourier-Jacobi models for real symplectic-metaplectic groups: the basic case

TL;DR

This work settles the basic tempered case of the local Gan-Gross-Prasad conjecture for Fourier-Jacobi models on real symplectic-metaplectic groups by combining the real-LLC framework with stable-range theta lifts and seesaw techniques. The authors prove multiplicity one and an epsilon-dichotomy across tempered L-packets, and establish the existence of Fourier-Jacobi models within each packet, leveraging the Shimura–Waldspurger correspondence and Jacobi–Weil representations. The approach navigates non-tempered reductions through stable-range arguments and a peeling-off strategy, building a bridge between tempered Fourier–Jacobi and Bessel models. Together with prior non-Archimedean and complex results, this work completes the archimedean instance of the conjecture and sharpens the LLC/Prasad-parameter dictionary for these dual pairs.

Abstract

In this paper, we generalize the method of Gan-Ichino and Atobe in [GI16][A18] to the field of real numbers and prove the basic tempered case of the local Gan-Gross-Prasad conjecture for Fourier-Jacobi models of symplectic-metaplectic groups, based on the tempered case of the conjecture for Bessel models proved in [CL22] by Chen-Luo.
Paper Structure (34 sections, 18 theorems, 196 equations)

This paper contains 34 sections, 18 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Local Langlands correspondence
  3. Weil groups and L-parameters
  4. Local Langlands correspondence for classical groups
  5. The symplectic group
  6. The odd special orthogonal group
  7. The even special orthogonal group
  8. Local Langlands correspondence for metaplectic groups
  9. Local Gan-Gross-Prasad Conjecture
  10. Symplectic-metaplectic case
  11. Distinguished character
  12. Statement of the conjecture
  13. Special orthogonal case
  14. Distinguished character
  15. Statement of Conjecture
  16. ...and 19 more sections

Key Result

Theorem 1.0.1

The following statements hold:

Theorems & Definitions (28)

  • Theorem 1.0.1
  • Theorem 3.1.1
  • Theorem 3.2.1
  • Proposition 4.1.1
  • Corollary 4.1.2
  • proof
  • Proposition 4.1.3
  • Proposition 4.2.1
  • Proposition 4.2.2
  • proof
  • ...and 18 more