[Paper Review] Unitary Friedberg-Jacquet periods

Rui Chen, Wee Teck Gan|arXiv (Cornell University)|Aug 9, 2021

Advanced Algebra and Geometry21 references4 citations

TL;DR

This paper studies unitary Friedberg-Jacquet periods on unitary groups via theta correspondence, establishing that the dimension of the Hom space for discrete series representations is at most one and determining its nonvanishing in terms of L-functions. It extends the classical GL(2n) results to the unitary setting, proving a unitary analogue of Theorem 1.1 and a partial global conjecture on central L-values and period nonvanishing.

ABSTRACT

The main goal of this paper is to study the unitary Friedberg-Jacquet period through their connection with the unitary Shalika period. Locally we study the multiplicity of unitary Friedberg-Jacquet periods for discrete series. Globally we prove one direction of a conjecture of Xiao-Zhang, stating that the non-vanishing of a global unitary Friedberg-Jacquet period implies the non-vanishing of the central value of the (twisted) standard L-function.

Motivation & Objective

• To extend the classical Friedberg-Jacquet period results for GL(2n) to the unitary group setting.
• To determine the dimension of unitary Friedberg-Jacquet periods for discrete series representations of U_{2n}(F) using theta correspondence.
• To establish a global implication: nonvanishing of the unitary FJ period implies nonvanishing of the central value of the twisted standard L-function.
• To propose a conjectural framework for unitary analogues of mirabolic Eisenstein series using Weil representations.

Proposed method

• Utilize theta correspondence to relate unitary Friedberg-Jacquet periods to unitary Shalika periods.
• Apply the local theory of Weil representations associated to skew-Hermitian spaces and conjugate-symplectic characters.
• Use the structure of the Weil representation ω_{W,μ,ψ} on unitary groups to define and analyze the Hom spaces for Shalika and FJ periods.
• Leverage known results on the unitary Shalika period from Beuzart-Plessis and Wan to deduce properties of the FJ period via duality.
• Apply the relative trace formula and endoscopic methods, drawing on work of Xiao-Zhang and Leslie.
• Use global zeta integrals and unfolding techniques to relate period integrals to special values of L-functions.

Experimental results

Research questions

• RQ1 For an irreducible discrete series representation π of U_{2n}(F), what is the dimension of Hom_{U_n × U_n}(π, χ₁ ⊗ χ₂) for unitary characters χ₁, χ₂?
• RQ2 Under what conditions does the unitary Friedberg-Jacquet period on U_{2n}(F) vanish or not vanish?
• RQ3 How does the nonvanishing of the global unitary FJ period relate to the central value of the standard L-function?
• RQ4 Can the unitary analog of the mirabolic Eisenstein series be constructed via Weil representations of U_n?
• RQ5 To what extent do the local Hom spaces Sha_B(π, ω_{B,μ,ψ}) and Lin_{W₀}(π, ω_{W₀,μ,ψ}) have dimension ≤1?

Key findings

• The dimension of Hom_{U_n × U_n}(π, χ₁ ⊗ χ₂) is at most one for any irreducible discrete series representation π of U_{2n}(F) and unitary characters χ₁, χ₂.
• The nonvanishing of the unitary Friedberg-Jacquet period for π is characterized by the nonvanishing of the standard L-function L(π, 1/2) and a pole of the exterior square L-function L(s, π, ∧²) at s = 1.
• The global unitary FJ period is nonvanishing only if the twisted standard L-function L(Π, 1/2) ≠ 0, supporting a partial global conjecture of Xiao-Zhang.
• The local Hom space Lin_{W₀}(π, ω_{W₀,μ,ψ}) for the unitary FJ period is shown to be at most one-dimensional via theta correspondence.
• The paper proposes that the unitary analog of the mirabolic Eisenstein series should be constructed from the Weil representation of U_n, generalizing the GL(V) case via ω_{W,μ,ψ}.
• The results extend the classical GL(2n) results (Theorem 1.1) to the unitary group setting, establishing a unitary analogue for discrete series representations.