La variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes linéaires

Abstract

We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated a la Arthur multiplied by the absolute value of the determinant to the power $s\in \mathbb{C}$. It has a geometric side which is a sum of distributions $I_{\mathfrak{o}}(s,\cdot )$ indexed by the invariants of the adjoint action of ${\mathrm{GL}}_n(\mathrm{F})$ on ${\mathfrak{gl}}_{n+1}(\mathrm{F})$ as well as a "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $I_{\mathfrak{o}}(s,\cdot )$ are invariant and depend only on the choice of the Haar measure on ${\mathrm{GL}}_n(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $I_{\mathfrak{o}}(s,\cdot )$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $I_{\mathfrak{o}}(s,\cdot )$ in terms of relative orbital integrals regularised by means of zeta functions.

42 pages, in French

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• Automorphic period and the central value of Rankin--Selberg L-function
• On the Arithmetic Fundamental Lemma in the minuscule case
• On the arithmetic transfer conjecture for exotic smooth formal moduli spaces
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Cited by in corpus (2)

• The Schwartz space of a smooth semi-algebraic stack
• An infinitesimal variant of Guo-Jacquet trace formula I: the case of $(G{L}_{2n,D},G{L}_{n,D}\times G{L}_{n,D})$