[Paper Review] Endoscopic transfer for unitary Lie algebras

Xiao Jingwei|arXiv (Cornell University)|Feb 21, 2018

Advanced Algebra and Geometry6 references3 citations

TL;DR

This paper provides a new local proof of the endoscopic transfer for unitary Lie algebras over non-archimedean local fields of characteristic zero, establishing its compatibility with Fourier transforms without relying on the fundamental lemma. The key result is a nilpotent orbit integral identity in the Jacquet-Rallis setting that reduces endoscopic transfers to general linear side identities via parabolic induction, thereby offering an alternative proof of the endoscopic fundamental lemma for unitary groups.

ABSTRACT

We give another proof of the existence of the endoscopic transfer for unitary Lie algebras and its compatibility with Fourier transforms. By the work of Kazhdan and Vashavsky, this implies the corresponding endoscopic fundamental lemma (theorem of Laumon--Ngô). We study the compatibility between Fourier transforms and transfers and we prove that the compatibility in the Jacquet-Rallis setting implies the compatibility in the endoscopic setting for unitary groups.

Motivation & Objective

• To provide an alternative proof of the endoscopic transfer for unitary Lie algebras without invoking the fundamental lemma.
• To establish the compatibility of endoscopic transfer with Fourier transforms using the Jacquet-Rallis transfer as a foundational tool.
• To demonstrate that the Jacquet-Rallis fundamental lemma implies the endoscopic fundamental lemma in the unitary case.
• To show that the transfer process commutes with Fourier transforms by verifying cancellation of sign factors.

Proposed method

• The proof relies on a key identity between nilpotent orbit integrals in the Jacquet-Rallis setting, relating orbit integrals on $\mathfrak{gl}(V) \times V \times V^*$ to sums over conjugacy classes in stable conjugacy classes via characters on $H^1(F, T_\gamma)$.
• The method uses germ expansion principles to extract nilpotent orbit integrals as limits of regular semisimple orbit integrals, leveraging the matching of functions under the Jacquet-Rallis transfer.
• Parabolic descent is applied to reduce the problem to lower-dimensional cases, enabling inductive control over the transfer structure.
• The transfer factor is computed explicitly by combining the Jacquet-Rallis transfer factors and the parabolic descent factor, showing their product equals $\chi(D(\delta))|D(\delta)|_F$.
• The construction is shown to commute with Fourier transforms by verifying that the $(-1)^{n-1}$ sign factor in the Fourier transform compatibility cancels precisely.
• The proof is purely local and avoids reduction to positive characteristic, distinguishing it from prior approaches.

Experimental results

Research questions

• RQ1 Does the endoscopic transfer for unitary Lie algebras exist and commute with Fourier transforms without assuming the fundamental lemma?
• RQ2 Can the endoscopic fundamental lemma for unitary groups be derived from the Jacquet-Rallis fundamental lemma via a local method?
• RQ3 How do nilpotent orbit integrals in the Jacquet-Rallis setting relate to those in the endoscopic setting?
• RQ4 What is the precise structure of the transfer factor in the endoscopic transfer for unitary Lie algebras?
• RQ5 Can the compatibility of Fourier transforms with endoscopic transfer be established through character sums over Galois cohomology classes?

Key findings

• The paper establishes the existence of the endoscopic transfer for unitary Lie algebras using only the Jacquet-Rallis transfer and parabolic descent, without assuming the fundamental lemma.
• It proves that the endoscopic transfer is compatible with Fourier transforms, with the compatibility condition holding due to cancellation of the $(-1)^{n-1}$ sign factor.
• A central identity (Theorem 4.7) reduces $\kappa$-orbit integrals in the endoscopic setting to a single nilpotent orbit integral on the general linear side.
• The transfer factor in the endoscopic transfer is shown to be the product of the Jacquet-Rallis transfer factors and the parabolic descent factor, yielding $\chi(D(\delta))|D(\delta)|_F$.
• The proof demonstrates that the Jacquet-Rallis fundamental lemma implies the endoscopic fundamental lemma in the unitary case, providing a new derivation of this result.
• The method provides a genuinely different, purely local proof of the endoscopic fundamental lemma, independent of Kazhdan-Vasavsky’s approach.