Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups

Abstract

We consider several questions about restriction of representations of classical and metaplectic groups over local and global fields to subgroups, extending considerably the scope of the earlier work on $SO(n),SO(n-1)$. This includes Bessel and Fourier-Jacobi models too. We formulate several conjectures about these restriction problems involving root numbers of symplectic representations in the local case, and central critical L-value in the global case. Along the way we prove several results both in number theory and representation theory.

112 pages

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Cited by in corpus (114)

• Towards a theory of local Shimura varieties
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• On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
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• A `relative' local Langlands correspondence
• Periods and nonvanishing of central L-values for GL(2n)
• Une formule intégrale reliée à la conjecture locale de Gross-Prasad, 2ème partie: extension aux représentations tempérées
• The orbit method and analysis of automorphic forms
• The Schwartz space of a smooth semi-algebraic stack
• On a lifting problem of L-packets
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• Hirzebruch-Zagier cycles and twisted triple product Selmer groups
• On the smooth transfer for Guo-Jacquet relative trace formulae
• Arthur Parameters and Cuspidal Automorphic Modules of Classical Groups
• La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires
• Arithmeticity for periods of automorphic forms
• Fine Deligne-Lusztig varieties and Arithmetic Fundamental Lemmas
• Symmetry breaking for orthogonal groups and a conjecture by B. Gross and D. Prasad
• The local theta correspondence and the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case
• A local converse Theorem for U(2,2)
• On Fourier coefficients of certain residual representations of symplectic groups
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• The Bessel Period Functional on SO(5): The Nontempered Case
• Isolation of cuspidal spectrum, with application to the Gan--Gross--Prasad conjecture
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• La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes linéaires
• Hecke and Galois Properties of Special Cycles on Unitary Shimura Varieties
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• Multiplicity one theorems for Fourier-Jacobi models
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• Arthur's multiplicity formula for even orthogonal and unitary groups
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• On Shimura varieties for unitary groups
• On interactions between harmonic analysis and the theory of automorphic forms
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• Hodge classes and the Jacquet-Langlands correspondence
• Descents of unipotent cuspidal representations of finite classical groups
• A vanishing Ext-branching theorem for $({\mathrm{GL}}_{n+1}(F),{\mathrm{GL}}_n(F))$
• Deligne's conjecture for automorphic motives over CM-fields
• Counting Automorphic Forms on Norm One Tori
• A New Proof to the Period Problems of GL(2)
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• Affine Hecke algebras for classical p-adic groups
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• Notes on MVW-extensions
• Generalised Whittaker models as instances of relative Langlands duality
• Galois representations for even general special orthogonal groups
• Bernstein-Zelevinsky derivatives, branching rules and Hecke algebras
• Standard Conjectures and Height Pairings
• The Bessel Period of U(3) and U(2) involving a non-tempered representation
• Endoscopic character identities for metaplectic groups
• The Prasad conjectures for U(2), SO(4) and Sp(4)
• Special L-values and Selmer groups of Siegel modular forms of genus 2
• Arthur Parameters and Fourier coefficients for Automorphic Forms on Symplectic Groups
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• Parameters of Hecke algebras for Bernstein components of p-adic groups
• Periods of Automorphic Forms Associated to Strongly Tempered Spherical Varieties
• Fourier-Jacobi cycles and arithmetic relative trace formula (with an appendix by Chao Li and Yihang Zhu)
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• Local Langlands Correspondence for Unitary Groups via Theta Lifts
• On the Gan-Gross-Prasad problem for finite classical groups
• Uniqueness of Ginzburg-Rallis models: the Archimedean case
• Descents of unipotent representations of finite unitary groups
• The Gross-Zagier-Zhang formula over function fields
• On the global Gan-Gross-Prasad conjecture for general spin groups
• Theta Correspondence and Arthur packets
• An arithmetic intersection conjecture
• Depth preserving property of the local Langlands correspondence for non-quasi-split unitary groups
• Special values of $L$-functions and the refined Gan-Gross-Prasad conjecture
• A local converse theorem for ${\text{Sp}}_{2r}$
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• A relation between multiplicity one and Bocherer's conjecture
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• Non-vanishing of Miyawaki type lift
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• On Miyawaki lifts with respect to Hermitian Maass lifts
• Distinction and Base Change
• Uniqueness of Rankin-Selberg periods
• Vanishing of quasi-invariant generalized functions
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