La formule des traces stable pour le groupe m\'etaplectique: les termes elliptiques

TL;DR

This paper stabilizes the elliptic semi-simple terms of the trace formula for the metaplectic group over a number field, expressing them via stable orbital integrals and providing explicit coefficient formulas.

Contribution

It extends the stabilization of the trace formula to the metaplectic group, using transfer of test functions and the fundamental lemma, with explicit formulas for coefficients.

Findings

Expression in terms of stable orbital integrals for certain elliptic classes

Explicit formulas for the coefficients in the stabilized trace formula

Verification of transfer and fundamental lemma in this context

Abstract

Let F be a number field and $\widetilde{\mathrm{Sp}}(2n)$ be the metaplectic covering of Weil of $\mathrm{Sp}(2n, \mathbb{A}_F)$. We stabilize the elliptic semi-simple terms of the genuine part of trace formula for $\widetilde{\mathrm{Sp}}(2n)$, by using the transfer of test functions and the fundamental lemma established in a previous article. We obtain therefore an expression in terms of stable orbital integrals along certain $F$-elliptic classes in $\mathrm{SO}(2n'+1) \times \mathrm{SO}(2n''+1)$, which we call equisingular, for pairs $(n', n'')$ satisfying $n'+n''=n$. A simple formula for the coefficients is also obtained.