[Paper Review] A Local Relative Trace Formula for the Ginzburg-Rallis Model: the Geometric Side

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

Advanced Algebra and Geometry12 references3 citations

TL;DR

This paper establishes the geometric side of a local relative trace formula for the Ginzburg-Rallis model over p-adic fields, using Waldspurger and Beuzart-Plessis' method. It derives a multiplicity formula for supercuspidal representations and proves multiplicity one for Vogan packets in this setting, advancing the local Gan-Gross-Prasad-type conjectures for the GL₆ exterior cube L-function.

ABSTRACT

Following the method developed by Waldspurger and Beuzart-Plessis in their proofs of the local Gan-Gross-Prasad conjecture, we are able to prove the geometric side of a local relative trace formula for the Ginzburg-Rallis model. Then by applying such formula, we prove a multiplicity formula of the Ginzburg-Rallis model for the supercuspidal representations. Using that multiplicity formula, we prove the multiplicity one theorem for the Ginzburg-Rallis model over Vogan packets in the supercuspidal case.

Motivation & Objective

• To establish the geometric side of a local relative trace formula for the Ginzburg-Rallis model over p-adic fields.
• To derive a multiplicity formula for irreducible supercuspidal representations of the Levi subgroup in the Ginzburg-Rallis model.
• To prove the multiplicity one theorem for Vogan packets in the supercuspidal case for this model.
• To extend techniques from the local Gan-Gross-Prasad conjecture to the Ginzburg-Rallis setting via relative trace formula methods.

Proposed method

• Applies the method of Waldspurger and Beuzart-Plessis, using weighted orbital integrals and Shalika germs to analyze the geometric side of the trace formula.
• Employs localization techniques for strongly cuspidal functions on the Lie algebra and group level, focusing on semisimple elements and their neighborhoods.
• Uses integral transfer and premier transformations to relate orbital integrals across different realizations of the model.
• Applies a limit process as N → ∞ to truncated functions, computing the limit of the trace formula integral.
• Relies on the structure of parabolic subgroups and the reduced model construction to analyze the geometric side in various cases.
• Establishes the trace formula via a principal proposition on convergence and combinatorial definitions of orbital integrals.

Experimental results

Research questions

• RQ1 What is the geometric side of the local relative trace formula for the Ginzburg-Rallis model?
• RQ2 How can the multiplicity of a supercuspidal representation in the Ginzburg-Rallis model be computed?
• RQ3 Does the multiplicity one property hold for Vogan packets in the supercuspidal case for this model?
• RQ4 How do the orbital integrals and Shalika germs behave under localization and truncation?
• RQ5 What is the role of the reduced model in simplifying the trace formula for non-maximal parabolic subgroups?

Key findings

• The geometric side of the local relative trace formula is fully established for the Ginzburg-Rallis model using localization and orbital integral techniques.
• A multiplicity formula is derived, showing that for supercuspidal representations, the multiplicity $ m( heta_ ho) = c_{ heta_ ho, ext{reg}}(1) = 1 $.
• The multiplicity one theorem holds for Vogan packets in the supercuspidal case, confirming that each such packet contributes at most one copy to the model.
• For type II models (e.g., (5,1), (3,3), (1,5) parabolics), the geometric side of the trace formula reduces to the germ at the identity, due to all semisimple elements being split.
• The limit $ oxed{ extstylerac{1}{2}} $ of the truncated integral $ oxed{ extstylerac{1}{2}} $ as $ N o oxed{ extstylerac{1}{2}} $ is shown to equal the orbital integral coefficient $ c_{ heta_f, ext{reg}}(1) $, confirming the trace formula.
• The trace formula and multiplicity formula are extended to the reduced model of maximal parabolic subgroups, with consistent results across types.