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Refined global Gan–Gross–Prasad conjecture for Fourier–Jacobi periods on symplectic groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  19 January 2017

Hang Xue
Hang Xue*
Affiliation:
School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA email [email protected]
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Abstract

In this paper, we propose a conjectural identity between the Fourier–Jacobi periods on symplectic groups and the central value of certain Rankin–Selberg $L$-functions. This identity can be viewed as a refinement to the global Gan–Gross–Prasad conjecture for $\text{Sp}(2n)\times \text{Mp}(2m)$. To support this conjectural identity, we show that when $n=m$ and $n=m\pm 1$, it can be deduced from the Ichino–Ikeda conjecture in some cases via theta correspondences. As a corollary, the conjectural identity holds when $n=m=1$ or when $n=2$, $m=1$ and the automorphic representation on the bigger group is endoscopic.

Keywords

Gan–Gross–Prasad conjecturesFourier–Jacobi periods

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 1 , January 2017 , pp. 68 - 131
Copyright
© The Author 2017 

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