Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T17:03:43.622Z Has data issue: false hasContentIssue false
  • English
  • Français

COMPARISON OF LOCAL RELATIVE CHARACTERS AND THE ICHINO–IKEDA CONJECTURE FOR UNITARY GROUPS

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  22 January 2020

Raphaël Beuzart-Plessis Open the ORCID record for Raphaël Beuzart-Plessis [Opens in a new window]
Raphaël Beuzart-Plessis*
Affiliation:
Université d’Aix-Marseille, I2M-CNRS(UMR 7373), Campus de Luminy, 13288Marseille Cédex 9, France (rbeuzart@gmail.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

Keywords

relative trace formularelative charactersIchino–Ikeda conjectureautomorphic representationsunitary groups

MSC classification

Primary: 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
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 6 , November 2021 , pp. 1803 - 1854
Check for updates
Copyright
© Cambridge University Press 2020

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., Gourevitch, D., Rallis, S. and Schiffmann, G., Multiplicity one theorems, Ann. of Math. (2) 172(2) (2010), 1407–1434.CrossRefGoogle Scholar
Aubert, A.-M., Moussaoui, A. and Solleveld, M., Generalizations of the Springer correspondence and cuspidal Langlands parameters, Manuscripta Math. 157(1–2) (2018), 121–192.CrossRefGoogle Scholar
Bergeron, N. and Clozel, L., Spectre automorphe des variétés hyperboliques et applications topologiques, Astérisque 303 (2005), xx+218 pp.Google Scholar
Bernstein, J. and Deligne, P., Le ‘centre’ de Bernstein, in Representations des groupes réductifs sur un corps local, Travaux en cours. (ed. Deligne, P.), pp. 1–32 (Hermann, Paris, 1984).Google Scholar
Bernstein, J. N. and Krötz, B., Smooth Fréchet globalizations of Harish-Chandra modules, Israel J. Math. 199(1) (2014), 45–111.CrossRefGoogle Scholar
Beuzart-Plessis, R., A local trace formula for the Gan–Gross–Prasad conjecture for unitary groups: the Archimedean case, Astérisque, to appear, arXiv:1506.01452.Google Scholar
Beuzart-Plessis, R., Endoscopie et conjecture locale raffinée de Gan–Gross–Prasad pour les groupes unitaires, Compos. Math. 151(7) (2015), 1309–1371.CrossRefGoogle Scholar
Beuzart-Plessis, R., La conjecture locale de Gross–Prasad pour les représentations tempérées des groupes unitaires, Mém. Soc. Math. Fr. (N.S.) (149) (2016), vii+191 pp.Google Scholar
Borel, A. and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535.CrossRefGoogle Scholar
Casselman, W., Canonical extensions of Harish-Chandra modules to representations of G , Canad. J. Math. 41(3) (1989), 385–438.CrossRefGoogle Scholar
Chaudouard, P.-H., On relative trace formulae: the case of Jacquet–Rallis, Acta Math. Vietnam. 44(2) (2019), 391–430.10.1007/s40306-018-00312-3CrossRefGoogle Scholar
Clozel, L., Orbital integrals on p-adic groups: a proof of the Howe conjecture, Ann. of Math. (2) 129(2) (1989), 237–251.CrossRefGoogle Scholar
Clozel, L., Characters of nonconnected, reductive p-adic groups, Canad. J. Math. 39(1) (1987), 149–167.CrossRefGoogle Scholar
Deligne, P., Le support du caractère d’une représentation supercuspidale, C. R. Acad. Sci. Paris A-B 283(4, Aii) (1976), A155–A157.Google Scholar
Dixmier, J. and Malliavin, P., Factorisations de fonctions et de vecteurs indéfiniment différentiables, Bull. Sci. Math. (2) 102(4) (1978), 307–330.Google Scholar
Gan, W. T., Gross, B. H. 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 No. 346, pp. 1–109 (Société Mathématique de France, Paris, 2012).Google Scholar
Gan, W. T. and Ichino, A., The Gross–Prasad conjecture and local theta correspondence, Invent. Math. 206(3) (2016), 705–799.CrossRefGoogle Scholar
Neal Harris, R., The refined Gross–Prasad conjecture for unitary groups, Int. Math. Res. Not. IMRN 2014(2) (2014), 303–389.CrossRefGoogle Scholar
Harris, M. and Taylor, R., The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, Volume 151 (Princeton University Press, Princeton, NJ, 2001). With an appendix by Vladimir G. Berkovich.Google Scholar
Henniart, G., Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique, Invent. Math. 139(2) (2000), 439–455.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(5) (2010), 1378–1425.CrossRefGoogle Scholar
Ichino, A., Lapid, E. and Mao, Z., On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups, Duke Math. J. 166(7) (2017), 1301–1348.10.1215/00127094-0000001XCrossRefGoogle Scholar
Ichino, A. and Zhang, W., Spherical characters for a strongly tempered pair, appendix to [57], Ann. of Math. (2) 180(3) (2014), 1033–1037.Google Scholar
Jacquet, H., Archimedean Rankin–Selberg integrals, in Automorphic Forms and L-Functions II. Local Sspects, Contemporary Mathematics, Volume 489, pp. 57–172 (American Mathematical Society, Providence, RI, 2009).Google Scholar
Jacquet, H., Piatetski-Shapiro, I. I. and Shalika, J. A., Rankin–Selberg convolutions, Amer. J. Math. 105(2) (1983), 367–464.CrossRefGoogle Scholar
Jacquet, H. and Rallis, S., On the Gross–Prasad conjecture for unitary groups, in On Certain L-Functions, Clay Mathematics Proceedings, Volume 13, pp. 205–264 (American Mathematical Society, Providence, RI, 2011).Google Scholar
Jacquet, H. and Shalika, J. A., On Euler products and the classification of automorphic representations I, Amer. J. Math. 103(3) (1981), 499–558.CrossRefGoogle Scholar
Kaletha, T., Minguez, A., Shin, S. W. and White, P.-J., Endoscopic classification of representations: inner forms of unitary groups, prepublication, 2014, arXiv:1409.3731.Google Scholar
Kottwitz, R. E., Harmonic analysis on reductive p-adic groups and Lie algebras, in Harmonic Analysis, the Trace Formula, and Shimura Varieties, Clay Mathematics Proceedings, Volume 4, pp. 393–522 (American Mathematical Society, Providence, RI, 2005).Google Scholar
Langlands, R. P., On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie Groups, Mathematical Surveys and Monographs, Volume 31, pp. 101–170 (American Mathematical Society, Providence, RI, 1989).CrossRefGoogle Scholar
Luna, D., Slices étalés, Mém. Soc. Math. Fr. 33 (1973), 81–105.Google Scholar
Luo, W., Rudnick, Z. and Sarnak, P., On the generalized Ramanujan conjecture for GL(n), in Automorphic Forms, Automorphic Representations, and Arithmetic (Fort Worth, TX, 1996), Proceedings of Symposia in Pure Mathematics, Volume 66, pp. 301–310 (American Mathematical Society, Providence, RI, 1999).CrossRefGoogle Scholar
Mœglin, C., Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité, J. Eur. Math. Soc. (JEMS) 4(2) (2002), 143–200.CrossRefGoogle Scholar
Mœglin, C., Classification et changement de base pour les séries discrètes des groupes unitaires p-adiques, Pacific J. Math. 233(1) (2007), 159–204.CrossRefGoogle Scholar
Mœglin, C. and Tadìc, M., Construction of discrete series for classical p-adic groups, J. Amer. Math. Soc. 15(3) (2002), 715–786.CrossRefGoogle Scholar
Mœglin, C. and Waldspurger, J.-L., La formule des traces locale tordue, Mem. Amer. Math. Soc. 251(1198) (2018), v+183 pp.Google Scholar
Mœglin, C. and Waldspurger, J.-L., Spectral decomposition and Eisenstein series. Une paraphrase de l’Écriture, Cambridge Tracts in Mathematics, Volume 113, p. xxviii+338 pp. (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
Mok, C. P., Endoscopic Classification of representations of Quasi-Split Unitary Groups, Mem. Amer. Math. Soc. 235(1108) (2015), 250 pages.Google Scholar
Moussaoui, A., Centre de Bernstein dual pour les groupes classiques, Represent. Theory 21 (2017), 172–246.CrossRefGoogle Scholar
Muìc, G., Some results on square integrable representations; irreducibility of standard representations, Int. Math. Res. Not. IMRN (14) (1998), 705–726.10.1155/S1073792898000427CrossRefGoogle Scholar
Müller, W., The trace class conjecture without the K-finiteness assumption, C. R. Acad. Sci. Paris Sér. I Math. 324(12) (1997), 1333–1338.CrossRefGoogle Scholar
Müller, W. and Speh, B., Absolute convergence of the spectral side of the Arthur trace formula for GLn , Geom. Funct. Anal. 14(1) (2004), 58–93. With an appendix by E. M. Lapid.CrossRefGoogle Scholar
Ramakrishnan, D., A theorem on $\text{GL}(n)$ a la Tchebotarev, prepublication, 2018, arXiv:1806.08429.Google Scholar
Sakellaridis, Y. and Venkatesh, A., Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017), viii+360 pp.Google Scholar
Scholze, P., The local Langlands correspondence for GLn over p-adic fields, Invent. Math. 192(3) (2013), 663–715.CrossRefGoogle Scholar
Silberger, A., Introduction to Harmonic Analysis on Reductive p-adic Groups: Based on Lectures by Harish-Chandra at the Institute for Advanced Study, Mathematical Notes, Volume 23, pp. 1971–1973 (Princeton University Press, Princeton, N.J., 1979).Google Scholar
Silberger, A., Special representations of reductive p-adic groups are not integrable, Ann. of Math. (2) 111 (1980), 571–587.CrossRefGoogle Scholar
Sun, B. and Zhu, C.-B., Multiplicity one theorems: the Archimedean case, Ann. of Math. (2) 175 (2012), 23–44.CrossRefGoogle Scholar
Tadíc, M., Geometry of dual spaces of reductive groups (non-Archimedean case), J. Anal. Math. 51 (1988), 139–181.CrossRefGoogle Scholar
Waldspurger, J.-L., Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie, Compos. Math. 54(2) (1985), 173–242.Google Scholar
Wallach, N., Real Reductive Groups. II, Pure and Applied Mathematics, Volume 132-II, pp. xiv+454, (Academic Press, Inc., Boston, MA, 1992).Google Scholar
Xu, B., On the cuspidal support of discrete series for p-adic quasisplit Sp(N) and SO(N), Manuscripta Math. 154(3-4) (2017), 441–502.10.1007/s00229-017-0923-xCrossRefGoogle 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 (With an appendix by Julia Gordon), Duke Math. J. 156(2) (2011), 167–227.Google Scholar
Zhang, W., Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups, Ann. of Math. (2) 180(3) (2014), 971–1049.CrossRefGoogle Scholar
Zhang, W., Automorphic period and the central value of Rankin–Selberg L-function, J. Amer. Math. Soc. 27(2) (2014), 541–612.CrossRefGoogle Scholar
Zhang, W., On arithmetic fundamental lemmas, Invent. Math. 188(1) (2012), 197–252.CrossRefGoogle Scholar