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

Wee Teck Gan, Benedict H. Gross+1 more|ArXiv.org|Sep 16, 2009

Advanced Algebra and GeometryMathematics67 references301 citations

TL;DR

This paper proposes a refined conjectural framework linking symplectic local root numbers, central critical L-values, and multiplicity-one phenomena in the restriction of irreducible representations from classical groups to their subgroups. Using Langlands-Vogan parameterization and Vogan L-packets, it predicts that exactly one generic representation in each L-packet has a unique H-invariant functional, with the central L-value's non-vanishing tied to this uniqueness via root numbers and component group characters.

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.

Motivation & Objective

• To formulate precise local and global conjectures on the restriction of irreducible representations from classical groups to their subgroups, particularly focusing on multiplicity-one phenomena.
• To establish a connection between the non-vanishing of central critical L-values and the existence of unique H-invariant functionals in representations.
• To unify and generalize previous conjectures on Bessel and Fourier-Jacobi models using Langlands parameters and component group characters.
• To reduce the general restriction problem to basic cases (dim V - dim W = 0 or 1) via structural theorems on root numbers and L-packet geometry.
• To provide a cohomological and motivic interpretation of the central critical L-value via Chow groups and automorphic forms on Shimura-type varieties.

Proposed method

• Utilizes Langlands-Vogan parameterization to classify irreducible representations of classical and metaplectic groups via complex L-parameters of the Weil-Deligne group.
• Applies symplectic root numbers of L-parameters to construct distinguished characters of the component group of the centralizer of the L-parameter.
• Introduces a local conjecture stating that exactly one generic representation in each Vogan L-packet satisfies dim Hom_H(π ⊗ ν̄, ℂ) = 1, determined by the distinguished character.
• Employs Matsushima’s formula and cohomological techniques to relate automorphic cohomology of Shimura varieties to the multiplicity of representations.
• Reduces the general restriction problem to basic cases (dim V - dim W = 0 or 1) using theorems on uniqueness of Bessel and Fourier-Jacobi models over non-archimedean local fields.
• Uses global geometry of cycles in Shimura varieties to link the central critical L-value of an automorphic representation to the non-vanishing of a height pairing on Chow groups.

Experimental results

Research questions

• RQ1 Which irreducible representations of a classical group admit a unique H-invariant functional under restriction to a subgroup H?
• RQ2 How do symplectic local root numbers and component group characters determine the multiplicity-one property in Vogan L-packets?
• RQ3 What is the precise relationship between the non-vanishing of the first derivative of the central critical L-value and the existence of a non-zero H-invariant functional?
• RQ4 How can the restriction problem for classical groups be reduced to basic cases involving dim V - dim W = 0 or 1?
• RQ5 To what extent do cohomological and Chow-theoretic constructions on Shimura varieties reflect automorphic multiplicity and L-function behavior?

Key findings

• The paper proves that d(π) ≤ 1 in almost all cases for Bessel and Fourier-Jacobi models over non-archimedean local fields, reducing the problem to basic cases.
• It establishes that the irreducible representations of the metaplectic group are classified via those of odd special orthogonal groups, generalizing results of Kudla-Rallis.
• A generalization of Deligne’s formula for orthogonal root numbers is proven for conjugate-dual representations (Proposition 5.2).
• The L-parameters of classical and unitary groups are described in a simplified form, as stated in Theorem 8.1.
• The local conjecture is shown to be compatible across different choices of additive character ψ, with dependence patterns varying by case (orthogonal, hermitian, symplectic, skew-hermitian).
• A refined global conjecture (Conjecture 27.1) is proposed, stating that L′(π₀, R, 1/2) ≠ 0 if and only if π_f embeds into CH^{n-1}(Σ(G), ℱ) with multiplicity one and the height pairing against Σ(H) is non-zero.