[Paper Review] Multiplicity One Theorem for the Ginzburg-Rallis Model: the tempered case

Chen Wan|arXiv (Cornell University)|Aug 12, 2016

Advanced Algebra and Geometry13 references3 citations

TL;DR

This paper proves the multiplicity one theorem for the Ginzburg-Rallis model in the tempered case over p-adic and real local fields, establishing that the sum of multiplicities in the standard and quaternionic models equals one. Using Waldspurger’s and Beuzart-Plessis’s methods, it derives a geometric multiplicity formula via the local relative trace formula and links the multiplicity to the epsilon factor in the archimedean case.

ABSTRACT

Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local Gan-Gross-Prasad conjecture, we are able to prove the multiplicity one theorem for the Ginzburg-Rallis model over the Vogan packets in the tempered case. In some cases, we can also relate the multiplicity to the epsilon factor. This is a sequel of our work \cite{Wan15} in which we consider the supercuspidal case.

Motivation & Objective

• To establish the multiplicity one theorem for the Ginzburg-Rallis model in the tempered representation case over p-adic and real local fields.
• To verify Jiang’s conjecture that the sum of multiplicities in the split and quaternionic models equals one for tempered representations.
• To relate the multiplicity to the epsilon factor of the standard L-function in the archimedean case.
• To develop and apply a local relative trace formula for the Ginzburg-Rallis model to derive a geometric multiplicity formula.
• To extend the method of Waldspurger and Beuzart-Plessis to the Ginzburg-Rallis setting, building on prior work in the supercuspidal case.

Proposed method

• Utilizes Waldspurger’s method for the local Gan-Gross-Prasad conjecture and Beuzart-Plessis’s techniques for the unitary case.
• Applies a local relative trace formula for the Ginzburg-Rallis model, defined via the integral $ I(f) = \int_{H(F)\backslash G(F)} I(f,g) dg $, with $ I(f,x) = \int_{Z_H(F)\backslash H(F)} f(x^{-1}hx) \xi(h)\omega(h) dh $.
• Employs the concept of strongly cuspidal functions in $ C_c^\infty(Z_G(F)\backslash G(F), \chi^{-2}) $ to define the spectral side via weighted orbital integrals.
• Introduces a quasi-character $ \theta_f $ on $ G(F) $, whose germ expansion at regular semisimple elements $ t \in T_{reg}(F) $ gives the coefficient $ c_{\theta_f, \mathcal{O}_t}(t) $.
• Defines the geometric multiplicity $ m_{\text{geom}}(\pi) $ as a sum over tori $ T \in \mathcal{T} $, involving $ |W(H,T)|^{-1} \nu(T) \int_{Z_G(F)\backslash T(F)} c_\pi(t) D^H(t) \chi(\det(t))^{-1} dt $.
• Uses the Jacquet-Langlands correspondence to relate representations on $ \mathrm{GL}_6(F) $ and $ \mathrm{GL}_3(D) $, and proves $ m(\pi) + m(\pi_D) = 1 $ for tempered $ \pi $.

Experimental results

Research questions

• RQ1 Does the multiplicity of the Ginzburg-Rallis model vanish or equal one for tempered representations of $ \mathrm{GL}_6(F) $?
• RQ2 Is the sum of multiplicities in the split and quaternionic Ginzburg-Rallis models equal to one for tempered representations?
• RQ3 Can the multiplicity be expressed geometrically via a trace formula involving nilpotent orbits and tori?
• RQ4 How does the epsilon factor $ \epsilon(1/2, \pi, \wedge^3) $ relate to the multiplicity in the archimedean case?
• RQ5 Can the method used in the supercuspidal case be extended to the tempered case using Waldspurger’s and Beuzart-Plessis’s techniques?

Key findings

• The multiplicity $ m(\pi) $ of any irreducible tempered representation $ \pi $ of $ \mathrm{GL}_6(F) $ with central character $ \chi^2 $ satisfies $ m(\pi) \leq 1 $, and $ m(\pi) + m(\pi_D) = 1 $, where $ \pi_D $ is the Jacquet-Langlands transfer to $ \mathrm{GL}_3(D) $.
• For $ F = \mathbb{R} $, the multiplicity $ m(\pi) = 1 $ if and only if $ \epsilon(1/2, \pi, \wedge^3) = 1 $, and $ m(\pi) = 0 $ if and only if $ \epsilon(1/2, \pi, \wedge^3) = -1 $.
• For p-adic fields, the same epsilon factor condition determines the multiplicity: $ m(\pi) = 1 \iff \epsilon(1/2, \pi, \wedge^3) = 1 $, provided $ \pi $ is not a discrete series.
• The geometric multiplicity formula $ m(\pi) = m_{\text{geom}}(\pi) $ holds for tempered representations, with the sum over tori $ T \in \mathcal{T} $ and coefficients derived from germ expansions of quasi-characters.
• The trace formula integral $ I(f) $ is absolutely convergent for strongly cuspidal functions, ensuring the validity of the spectral and geometric side decomposition.
• The method successfully extends the multiplicity one result from the supercuspidal case to the tempered case, completing the verification of Jiang’s conjecture in this setting.