A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the archimedean case

TL;DR

This paper establishes a local trace formula for unitary groups over real fields, leading to a geometric expression for multiplicities in the Gan-Gross-Prasad conjecture and confirming a weak multiplicity one property in the archimedean case.

Contribution

It extends the local Gan-Gross-Prasad conjecture results to the archimedean case by deriving a local trace formula and geometric multiplicity formulas, previously known only for p-adic fields.

Findings

Derived a geometric formula for multiplicities in the conjecture

Proved a weak multiplicity one property for tempered L-packets over real fields

Extended known results from p-adic to archimedean fields

Abstract

In this paper, we prove, following earlier work of Waldspurger ([Wa1], [Wa4]), a sort of local relative trace formula which is related to the local Gan-Gross-Prasad conjecture for unitary groups over a local field $F$ of characteristic zero. As a consequence, we obtain a geometric formula for certain multiplicities $m(\pi)$ appearing in this conjecture and deduce from it a weak form of the local Gan-Gross-Prasad conjecture (multiplicity one in tempered L-packets). These results were already known over $p$-adic fields and thus are only new when $F=\mathbb{R}$.