Fourier transform and endoscopic transfer on real Lie algebras

Technical Summary: Fourier Transform and Endoscopic Transfer on Real Lie Algebras

Problem Statement
The paper addresses the relationship between the Fourier transform and endoscopic transfer on real Lie algebras. Specifically, it seeks to establish the archimedean analogue of a fundamental theorem by J.-L. Waldspurger ([Wal97]), which connects smooth transfer and the Fourier transform on reductive Lie algebras. While Waldspurger's original result for $p$-adic fields relies on global methods (the Lie algebra trace formula) and the Fundamental Lemma, and the complex case is known to be local, the real case required a proof using purely local methods. The central problem is to prove that the endoscopic transfer commutes with the Fourier transform over $\mathbb{R}$ without invoking global trace formulas.

Methodology
The authors, Cheng Chen and Zhilin Luo, employ a purely local approach to prove the compatibility of the Fourier transform with endoscopic transfer. Their methodology proceeds in three main stages:

1. Reduction to the Elliptic Case: Following the strategy of Waldspurger ([Wal97, §6]), the authors reduce the general identity to the case where the endoscopic datum is elliptic and the elements involved are elliptic regular semisimple. This reduction relies on the spectral characterization of orbital integrals and stable orbital integrals for test and Schwartz functions, drawing on the work of Shelstad ([She79a], [She79b]) and Bouaziz ([Bou94a], [Bou94b]).

2. Explicit Verification in the Elliptic Locus: For the elliptic case, the authors utilize a result by W. Rossmann ([Ros78]) regarding the explicit formula for the Fourier transform of orbital integrals on elliptic Cartan subalgebras. They compute the transfer factors ($\Delta$) and the distributions $D_{G,H}$ and $\tilde{D}_{G,H}$ explicitly. By analyzing the structure of the transfer factors ($\Delta_I, \Delta_{II}, \Delta_{III}$) and the behavior of the Weyl groups and root systems under the relevant isomorphisms, they verify the identity directly for elliptic elements.

3. Uniqueness via Harish-Chandra's Theorem: To extend the result from the elliptic locus to the entire Lie algebra, the authors invoke Harish-Chandra's uniqueness theorem ([Var77, Theorem 13]). They demonstrate that the distributions in question, when normalized by the square root of the Weyl discriminant, are:

   • Tempered distributions.
   • Invariant under the conjugation action of $G(\mathbb{R})$.
   • $I(\mathfrak{g})$-finite (eigendistributions for the algebra of invariant differential operators).
   • Of "elliptic type."
     Since such distributions are uniquely determined by their values on the elliptic locus, the identity proven in the elliptic case implies the identity holds globally.

Key Contributions and Results
The paper establishes the following main results:

• Theorem 1.0.1 (Smooth Transfer): Reaffirms the existence of smooth transfer for both test functions ($C^\infty_c$) and Schwartz functions ($S$) on real Lie algebras, following the spectral characterization established by Bouaziz and Shelstad.
• Theorem 1.0.3 (Waldspurger's Identity): Proves the identity $D_{G,H}(X_H, X_G) = \tilde{D}_{G,H}(X_H, X_G)$ for any $X_H$ in the $G^*$-regular locus and $X_G$ in the regular locus. Here, $D_{G,H}$ represents the Fourier transform of the endoscopic transfer, and $\tilde{D}_{G,H}$ represents the transfer of the Fourier transform (adjusted by Weil constants).
• Corollary 1.0.4 (Compatibility): As a direct consequence, the paper proves that the Fourier transform commutes with the endoscopic transfer. Specifically, if $f_G$ and $f_H$ satisfy the smooth transfer condition, then their Fourier transforms satisfy a corresponding relation involving the Weil constants $\gamma_\psi$.
• Proposition 1.0.5: The authors deduce that the Fourier transform of a stable distribution on a real Lie algebra is itself stable.

Significance and Claims
The authors claim that their proof is purely local, contrasting it with Waldspurger's proof for $p$-adic fields, which is global in nature and relies on the Fundamental Lemma for the unit element at almost all local places. The paper does not claim to prove the Fundamental Lemma itself; rather, it establishes the compatibility of the Fourier transform and endoscopic transfer as a standalone result in the archimedean setting.

The significance of this work lies in providing a self-contained, local proof for the real case, analogous to the known result for the complex case. The authors note that this result is expected by experts and serves as a necessary step for further studies on stable distributions supported on the nilpotent cone, following Waldspurger's work in the $p$-adic setting. They also suggest that while the weighted version of this theorem exists for $p$-adic fields (via Chaudouard and Laumon), establishing a weighted analogue for the real case using purely local methods remains an open and interesting direction for future investigation.