Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T09:36:14.208Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Lie groups Discontinuous groups and automorphic forms

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, Francepierre-henri.chaudouard@imj-prg.fr
Michał Zydor
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI48109, USAzydor@umich.edu
Get access Rights & Permissions [Opens in a new window]

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
Check for updates
Copyright
© The Author(s) 2021

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aizenbud, A. and Gourevitch, D., Schwartz functions on Nash manifolds, Int. Math. Res. Not. IMRN 2008 (2008), rmn155.10.1093/imrn/rnm155CrossRefGoogle Scholar
Aizenbud, A. and Gourevitch, D., Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet–Rallis's theorem, Duke Math. J. 149 (2009), 509–567. With an appendix by the authors and E. Sayag.CrossRefGoogle Scholar
Aizenbud, A. and Gourevitch, D., The de-Rham theorem and Shapiro lemma for Schwartz function on Nash manifolds, Israel J. Math. 177 (2010), 155–188.CrossRefGoogle Scholar
Aizenbud, A., A partial analog of the integrability theorem for distributions on $p$-adic spaces and applications, Israel J. Math. 193 (2013), 233–262.CrossRefGoogle Scholar
Arthur, J., The characters of discrete series as orbital integrals, Invent. Math. 32 (1976), 205–261.CrossRefGoogle Scholar
Arthur, J., A trace formula for reductive groups I. Terms associated to classes in $G(\mathbb {Q})$, Duke Math. J. 45 (1978), 911–952.CrossRefGoogle Scholar
Arthur, J., The trace formula in invariant form, Ann. of Math. (2) 114 (1981), 1–74.CrossRefGoogle Scholar
Arthur, J., On a family of distributions obtained from orbits, Canad. J. Math. 38 (1986), 179–214.CrossRefGoogle Scholar
Bochnak, J., Coste, M. and Roy, M.-F., Real algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 36 (Springer, Berlin, 1998). Translated from the 1987 French original, revised by the authors.CrossRefGoogle Scholar
Beuzart-Plessis, R., Comparison of local spherical characters and the Ichino–Ikeda conjecture for unitary groups, Preprint (2016), arXiv:1602.06538.Google Scholar
Beuzart-Plessis, R., Cours Peccot (2017).Google Scholar
Beuzart-Plessis, R., Plancherel formula for ${GL_n(F)\backslash GL_n(E)}$ and applications to the Ichino–Ikeda and formal degree conjectures for unitary groups, Preprint (2018), arXiv:1812.00047.Google Scholar
Beuzart-Plessis, R., A new proof of Jacquet–Rallis's fundamental lemma, Preprint (2019), arXiv:1901.02653.Google Scholar
Chaudouard, P.-H., La formule des traces pour les algèbres de Lie, Math. Ann. 322 (2002), 347–382.CrossRefGoogle Scholar
Chaudouard, P.-H., On relative trace formulae: the case of Jacquet–Rallis, Acta Math. Vietnam. 44 (2019), 391–430.CrossRefGoogle Scholar
du Cloux, F., Sur les représentations différentiables des groupes de Lie algébriques, Ann. Sci. Éc. Norm. Supér. (4) 24 (1991), 257–318.CrossRefGoogle Scholar
Gan, W. T., Gross, B. and Prasad, D., Symplectic local root numbers, central critical $L$ values, and restriction problems in the representation theory of classical groups, in Sur les conjectures de Gross et Prasad. I, Astérisque, vol. 346 (Société Mathématique de France, Paris, 2012), 1–109.Google Scholar
Harris, R. N., The refined Gross–Prasad conjecture for unitary groups, Int. Math. Res. Not. IMRN 2014 (2014), 303–389.CrossRefGoogle Scholar
Ichino, A. and Ikeda, T., On the periods of automorphic forms on special orthogonal groups and the Gross–Prasad conjecture, Geom. Funct. Anal. 19 (2010), 1378–1425.CrossRefGoogle Scholar
Jacquet, H., Sur un résultat de Waldspurger, Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), 185–229.CrossRefGoogle Scholar
Jacquet, H. and Rallis, S., On the Gross–Prasad conjecture for unitary groups, in On certain L-functions, Clay Mathematics Proceedings, vol. 13 (American Mathematical Society, Providence, RI, 2011), 205–265.Google Scholar
Kaletha, T., Minguez, A., Shin, S. W. and White, P.-J., Endoscopic classification of representations: inner forms of unitary groups, Preprint (2014), arXiv:1409.3731.Google Scholar
Luna, D. and Richardson, R. W., A generalization of the Chevalley restriction theorem, Duke Math. J. 46 (1979), 487–496.CrossRefGoogle Scholar
Labesse, J.-P. and Waldspurger, J.-L., La formule des traces tordue d'après le Friday Morning Seminar, CRM Monograph Series, vol. 31 (American Mathematical Society, Providence, RI, 2013). With a foreword by Robert Langlands [dual English/French text].CrossRefGoogle Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, third edition (Springer, Berlin, 1994).CrossRefGoogle Scholar
Mok, C. P., Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235(1108) (2015).Google Scholar
Rallis, S. and Schiffman, G., Multiplicity one conjectures, Preprint (2008), arXiv:0705.2168v1.Google Scholar
Serre, J.-P., Lie algebras and Lie groups, Lectures given at Harvard University, vol. 1964 (W. A. Benjamin, New York–Amsterdam, 1965).Google Scholar
Wallach, N., Real reductive groups. II, Pure and Applied Mathematics, vol. 132 (Academic Press, 1992).Google Scholar
Xue, H., On the global Gan–Gross–Prasad conjecture for unitary groups: approximating smooth transfer of Jacquet–Rallis, J. Reine Angew. Math. 756 (2019), 65–100.CrossRefGoogle Scholar
Yun, Z., The fundamental lemma of Jacquet and Rallis, Duke Math. J. 156 (2011), 167–227. With an appendix by Julia Gordon.Google Scholar
Zhang, W., On arithmetic fundamental lemmas, Invent. Math. 188 (2012), 197–252.CrossRefGoogle Scholar
Zhang, W., On the smooth transfer conjecture of Jacquet–Rallis for $n=3$, Ramanujan J. 29 (2012), 225–256.CrossRefGoogle Scholar
Zhang, W., Automorphic period and the central value of Rankin–Selberg $L$-function, J. Amer. Math. Soc. 27 (2014), 541–612.CrossRefGoogle Scholar
Zhang, W., Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. (2) 180 (2014), 971–1049.CrossRefGoogle Scholar
Zydor, M., La variante infinitésimale de la formule des traces de Jacquet–Rallis pour les groupes unitaires, Canad. J. Math. 68 (2016), 1382–1435.CrossRefGoogle Scholar
Zydor, M., La variante infinitésimale de la formule des traces de Jacquet–Rallis pour les groupes linéaires, J. Inst. Math. Jussieu 17 (2018), 735–783.CrossRefGoogle Scholar
Zydor, M., Les formules des traces relatives de Jacquet–Rallis grossières, J. Reine Angew. Math. 762 (2020), 195–259.CrossRefGoogle Scholar