La variante infinit\'esimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires

TL;DR

This paper develops an infinitesimal version of the Jacquet-Rallis trace formula for unitary groups, involving geometric and spectral sides, and introduces new invariant distributions related to orbital integrals.

Contribution

It establishes an infinitesimal trace formula for unitary groups, defining invariant distributions and expressing certain classes via regularized orbital integrals.

Findings

Distributions $J_{\mathfrak{o}}$ are invariant and depend only on Haar measure.

For regular semi-simple classes, $J_{\mathfrak{o}}$ are relative orbital integrals.

For relatively regular semi-simple classes, $J_{\mathfrak{o}}$ are expressed via regularized orbital integrals.

Abstract

We establish an infinitesimal version of the Jacquet-Rallis trace formula for unitary groups. Our formula is obtained by integrating a truncated kernel \`a la Arthur. It has a geometric side which is a sum of distributions $J_{\mathfrak{o}}$ indexed by classes of elements of the Lie algebra of $U(n+1)$ stable by $U(n)$-conjugation as well as the "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $J_{\mathfrak{o}}$ are invariant and depend only on the choice of the Haar measure on $U(n)(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $J_{\mathfrak{o}}$ in terms of relative orbital integrals regularised by means of z\^eta functions.