Multiplicity formula and stable trace formula

TL;DR

This paper stabilizes the local trace formula for reductive groups over $Q$, explicitly constructs the spectral side in the Archimedean case with cuspidal test functions, and derives multiplicity and Lefschetz number formulas.

Contribution

It provides the explicit spectral form of the stable local trace formula in the Archimedean case and establishes a new multiplicity formula for discrete series representations.

Findings

Explicit spectral side of stable local trace formula constructed.

Multiplicity formula for discrete series derived.

Stable $L^{2}$-Lefschetz number formula obtained.

Abstract

Let $G$ be a connected reductive group over $\mathbb{Q}$. In this paper, we will stabilize the local trace formula, in particular, we construct the explicit form of the spectral side of stable local trace formula in the Archimedean case, when one component of the test function is cuspidal. Then we will also give the multiplicity formula for discrete series. At the same time, we obtain the stable version of $L^{2}$-Lefschetz number formula.