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 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.

The Schwartz space of a smooth semi-algebraic stack

“…In preparation for the proof of Theorem 5.7.1, I present the crux of the argument, which is also at the heart of other methods of regularization of orbital integrals, cf. [48,46].…”

“…Example. The exponents are non-critical, and hence the relative trace formula can be defined by the methods of the current paper, for the quotient spaces (X ×X)/G diag when X is the homogeneous space of the Gross-Prasad conjectures (SO diag n \SO n × SO n+1 or U diag n \U n × U n+1 , considered in [48,47]), and for a variant of the corresponding Jacquet-Rallis relative trace formula for linear groups [46,47]. I expect that, in the unitary Gross-Prasad case that has been studied by Zydor, the resulting distribution coincides with his, but it would be an interesting exercise to show that.…”

“…However, there is a quadratic character entering the Jacquet-Rallis relative trace formula (on GL n or GL n+1 , whichever is even-dimensional), which I have presently not included in the formalism. However, the linearization close to the most singular point of X → c (again represented by the element 1) has been described in [46], and has to do with H := H n := GL n -orbital integrals, against a quadratic character, on the space of the representation V = {X ∈ h n+1,E |X +X = 0}. That is, the only difference from the setup of the present paper is that for the same pair (H, V ), one needs to put a quadratic character on H.…”

“…There is a very general approach to truncated orbital integrals by J. Levy [34] (who, however, does not produce invariant distributions), and several variants of the truncation method which are better suited for applications to the relative trace formula by Jacquet, Lapid, Rogawski, Ichino, Yamana, Zydor and others [28,26,48,46,47]. Joseph Bernstein sometimes expressed the wish to see this done without truncation, and this is what I do here, in a restricted setting that does not include the adjoint quotient of the group (but for reasons that cannot be addressed in a purely geometric way).…”

Le transfert singulier pour la formule des traces de Jacquet–Rallis

“…Le cas essentiel qui reste à traiter est la classe nilpotente correspondant à . On utilise alors un analogue dans la situation infinitésimale de la formule des traces de Jacquet–Rallis tel qu’établi par Zydor dans [Zyd16] et [Zyd18]. Le côté géométrique comprend toutes les distributions alors que le côté spectral fait apparaître leurs transformées de Fourier.…”