The spectral side of stable local trace formula for real groups

TL;DR

This paper constructs the spectral side of the stable local trace formula for real quasi-split groups directly, providing explicit expressions in terms of Langlands parameters, advancing the understanding of harmonic analysis on real groups.

Contribution

It explicitly constructs the spectral side of the stable local trace formula for quasi-split real groups, incorporating Shelstad's work and providing direct expressions in terms of Langlands parameters.

Findings

Explicit expression for the spectral side in terms of Langlands parameters

Construction of the spectral side directly for quasi-split K-groups over ℝ

Integration of Shelstad's work into the trace formula framework

Abstract

Let $G$ be a connected quasi-split reductive group over $\mathbb{R}$, and more generally, a quasi-split $K$-group over $\mathbb{R}$. Arthur had obtained the formal formula for the spectral side of the stable local trace formula, by using formal substitute of Langlands parameters. In this paper, we construct the spectral side of the stable trace formula and endoscopy trace formula directly for quasi-split $K$-groups over $\mathbb{R}$, by incorporating the works of Shelstad. In particular we give the explicit expression for the spectral side of the stable local trace formula, in terms of Langlands parameters.