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

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  16 March 2021

Pierre-Henri Chaudouard  and
Michał Zydor
Pierre-Henri Chaudouard
Affiliation:
Université de Paris et Sorbonne Université, CNRS, IMJ-PRG, F-75006Paris, [email protected]
Michał Zydor
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI48109, [email protected]
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Résumé

La formule des traces relative de Jacquet–Rallis (pour les groupes unitaires ou les groupes linéaires généraux) est une identité entre des périodes des représentations automorphes et des distributions géométriques. Selon Jacquet et Rallis, une comparaison de ces deux formules des traces relatives devrait aboutir à une démonstration des conjectures de Gan–Gross–Prasad et Ichino–Ikeda pour les groupes unitaires. Les termes géométriques des groupes unitaires ou des groupes linéaires sont indexés par les points rationnels d'un espace quotient commun. Nous établissons que ces termes géométriques peuvent être vus comme des fonctionnelles sur des espaces d'intégrales orbitales semi-simples régulières locales. En outre, nous montrons que point par point ces distributions sont en fait égales, via l'identification des espaces d'intégrales orbitales locales donnée par le transfert et le lemme fondamental (essentiellement connus dans cette situation). Cela donne leur comparaison et cela clôt la partie géométrique du programme de Jacquet–Rallis. Notre résultat principal est donc un analogue de la stabilisation de la partie géométrique de la formule des traces due à Langlands, Kottwitz et Arthur.

Abstract

Abstract

The relative trace formula of Jacquet–Rallis (for unitary groups or general linear groups) is an identity between periods of automorphic representations and geometric distributions. According to Jacquet and Rallis a comparison between these two trace formulae should give a proof of the Gan–Gross–Prasad and Ichino–Ikeda conjectures for unitary groups. The geometric terms are parametrized by the rational points of a common quotient space. In this paper, we show that the geometric terms both for unitary groups and general linear groups can be factorized as functionals on the space of local regular semi-simple orbital integrals. Moreover, we show for each rational point of the quotient that the corresponding terms are equal through the identification of the space of local orbital integrals given by the transfer and the fundamental lemma (essentially known in our situation). This gives the full comparison of the geometric sides and this achieves the geometric part of the Jacquet–Rallis program. Our main result is an analog of the geometric stabilization of the Arthur–Selberg trace formula due to Langlands, Kottwitz and Arthur.

Keywords

Langlands programGan–Gross–Prasad conjecturesIchino–Ikeda conjecturesJacquet–Rallis relative trace formulaorbital integrals

MSC classification

Primary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E50: Representations of Lie and linear algebraic groups over local fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 2 , February 2021 , pp. 303 - 434
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Copyright
© The Author(s) 2021

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