We establish an infinitesimal version of the Jacquet-Rallis trace formula for general linear groups. Our formula is obtained by integrating a kernel truncated à 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 $\text{GL}_{n}(\text{F})$ on $\mathfrak{gl}_{n+1}(\text{F})$ 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 $\text{GL}_{n}(\mathbb{A})$. 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.
Bounds for Fourier transforms of regular orbital integrals on $p$-adic Lie algebras