Geodesic
The global Gan-Gross-Prasad conjecture for unitary groups. II. From Eisenstein series to Bessel periods
Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e16

Voir la notice de l'article provenant de la source Cambridge University Press

We state and prove an extension of the global Gan-Gross-Prasad conjecture and the Ichino-Ikeda conjecture to the case of some Eisenstein series on unitary groups $U_n\times U_{n+1}$. Our theorems are based on a comparison of the Jacquet-Rallis trace formulas. A new point is the expression of some interesting spectral contributions in these formulas in terms of integrals of relative characters. As an application of our main theorems, we prove the global Gan-Gross-Prasad and the Ichino-Ikeda conjecture for Bessel periods of unitary groups. 
Beuzart-Plessis, Raphaël; Chaudouard, Pierre-Henri. The global Gan-Gross-Prasad conjecture for unitary groups. II. From Eisenstein series to Bessel periods. Forum of Mathematics, Pi, Tome 13 (2025) no. 1, p. e16. doi: 10.1017/fmp.2025.8
@article{10_1017_fmp_2025_8,
     author = {Beuzart-Plessis, Rapha\"el and Chaudouard, Pierre-Henri},
     title = {The global {Gan-Gross-Prasad} conjecture for unitary groups. {II.} {From} {Eisenstein} series to {Bessel} periods},
     journal = {Forum of Mathematics, Pi},
     pages = {e16},
     year = {2025},
     volume = {13},
     number = {1},
     doi = {10.1017/fmp.2025.8},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2025.8/}
}
TY  - JOUR
AU  - Beuzart-Plessis, Raphaël
AU  - Chaudouard, Pierre-Henri
TI  - The global Gan-Gross-Prasad conjecture for unitary groups. II. From Eisenstein series to Bessel periods
JO  - Forum of Mathematics, Pi
PY  - 2025
SP  - e16
VL  - 13
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2025.8/
DO  - 10.1017/fmp.2025.8
ID  - 10_1017_fmp_2025_8
ER  - 
%0 Journal Article
%A Beuzart-Plessis, Raphaël
%A Chaudouard, Pierre-Henri
%T The global Gan-Gross-Prasad conjecture for unitary groups. II. From Eisenstein series to Bessel periods
%J Forum of Mathematics, Pi
%D 2025
%P e16
%V 13
%N 1
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2025.8/
%R 10.1017/fmp.2025.8
%F 10_1017_fmp_2025_8

[AB68] Atiyah, M. and Bott, R., ‘A Lefschetz fixed point formula for elliptic complexes. II. Applications’, Ann. of Math. (2) 88 (1968), 451–491. Google Scholar | DOI

[ARS10] Aizenbud, D., Gourevitch, A., Rallis, S. and Schiffmann, G., ‘Multiplicity one theorems’, Ann. of Math. (2) 172(2) (2010), 1407–1434. Google Scholar | DOI

[Art78] Arthur, J., ‘A trace formula for reductive groups I. Terms associated to classes in ’, Duke Math. J. 45 (1978), 911–952. Google Scholar | DOI

[Art80] Arthur, J., ‘A trace formula for reductive groups II’, Comp. Math. 40 (1980), 87–121. Google Scholar

[Art81] Arthur, J., ‘The trace formula in invariant form’, Ann. of Math. (2) 114(1) (1981), 1–74. Google Scholar | DOI

[Art85] Arthur, J., ‘A measure on the unipotent variety’, Canad. J. Math. 37(6) (1985), 1237–1274. Google Scholar | DOI

[Beu21] Beuzart-Plessis, R., ‘Comparison of local spherical characters and the Ichino-Ikeda conjecture for unitary groups’, J. Inst. Math. Jussieu 20(6) (2021), 1803–1854. Google Scholar | DOI

[BLZZ21] Beuzart-Plessis, R., Liu, Y., Zhang, W. and Zhu, X., ‘Isolation of cuspidal spectrum, with application to the Gan-Gross-Prasad conjecture’, Ann. of Math. (2) 194(2) (2021), 519–584. Google Scholar | DOI

[BP18] Beuzart-Plessis, R., ‘Archimedean theory and -factors for the Asai Rankin-Selberg integrals’, Preprint, 2018, . Google Scholar | arXiv

[BP20] Beuzart-Plessis, R., ‘A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean case’, Astérisque 418 (2020), ix+305, Google Scholar

[BP21] Beuzart-Plessis, R., ‘Plancherel formula for and applications to the Ichino-Ikeda and formal degree conjectures for unitary groups’, Invent. Math. 225(1) (2021), 159–297. Google Scholar | DOI

[BPCZ22] Beuzart-Plessis, R., Chaudouard, P.-H. and Zydor, M., ‘The global Gan-Gross-Prasad conjecture for unitary groups: the endoscopic case’, Publ. Math. Inst. Hautes Études Sci. 135 (2022), 183–336. Google Scholar | DOI

[BR10] Badulescu, A. I. and Renard, D., ‘Unitary dual of at Archimedean places and global Jacquet-Langlands correspondence’, Compos. Math. 146(5) (2010), 1115–1164. Google Scholar | DOI

[Bum04] Bump, D., Lie Groups (Graduate Texts in Mathematics) vol. 225 (Springer-Verlag, New York, 2004). Google Scholar | DOI

[Cha22] Chaudouard, P.-H., ‘A spectral expansion for the symmetric space ’, Preprint, 2022, . Google Scholar | arXiv

[CHH88] Cowling, M., Haagerup, U. and Howe, R., ‘Almost matrix coefficients’, J. Reine Angew. Math. 387 (1988), 97–110. Google Scholar

[Cog08] Cogdell, J. W., ‘Notes on -functions for ’, in School on Automorphic Forms on , (ICTP Lect. Notes) vol. 21 (Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008), 75–158. Google Scholar

[CS80] Casselman, W. and Shalika, J., ‘The unramified principal series of -adic groups. II. The Whittaker function’, Compos. Math. 41(2) (1980), 207–231. Google Scholar

[CZ21] Chaudouard, P.-H. and Zydor, M., ‘Le transfert singulier pour la formule des traces de Jacquet-Rallis’, Compos. Math. 157(2) (2021), 303–434. Google Scholar | DOI

[Fli88] Flicker, Y., ‘Twisted tensors and Euler products’, Bull. Soc. Math. France 116(3) (1988), 295–313. Google Scholar | DOI

[Fli92] Flicker, Y., ‘Distinguished representations and a Fourier summation formula’, Bull. Soc. Math. France 120(4) (1992), 413–465. Google Scholar | DOI

[FLO12] Feigon, B., Lapid, E. and Offen, O., ‘On representations distinguished by unitary groups’, Publ. Math. Inst. Hautes Études Sci. 115 (2012), 185–323. Google Scholar | DOI

[Fra98] Franke, J., ‘Harmonic analysis in weighted -spaces’, Ann. Sci. École Norm. Sup. (4) 31(2) (1998), 181–279. Google Scholar | DOI

[GGP12] Gan, W. T., Gross, B. and Prasad, D., ‘Symplectic local root numbers, central critical values, and restriction problems in the representation theory of classical groups’, Astérisque 346 (2012), 1–109. Google Scholar

[GK72] Gelfand, I. M. and Každan, D. A., ‘Representations of the group where is a local field’, Funkcional. Anal. i Priložen. 6(4) (1972), 73–74. Google Scholar

[GK75] Gelfand, I. M. and Kajdan, D. A., ‘Representations of the group where is a local field’, in Lie Groups and Their Representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) (Halsted, New York, 1975), pages 95–118. Google Scholar

[Gro97] Gross, B., ‘On the motive of a reductive group’, Invent. Math. 130(2) (1997), 287–313. Google Scholar | DOI

[GRS11] Ginzburg, D., Rallis, S. and Soudry, D., The Descent Map from Automorphic Representations of to Classical Groups (World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011). Google Scholar

[Har14] Harris, R. N., ‘The refined Gross-Prasad conjecture for unitary groups’, Int. Math. Res. Not. IMRN 2 (2014), 303–389. Google Scholar | DOI

[II10] 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. Google Scholar | DOI

[IY15] Ichino, A. and Yamana, S., ‘Periods of automorphic forms: the case of ’, Compos. Math., 151(4) (2015), 665–712. Google Scholar | DOI

[IY19] Ichino, A. and Yamana, S., ‘Periods of automorphic forms: the case of ’, J. Reine Angew. Math. 746 (2019), 1–38. Google Scholar | DOI

[Jac04] Jacquet, H., ‘Integral representation of Whittaker functions’, in Contributions to Automorphic Forms, Geometry, and Number Theory (Johns Hopkins Univ. Press, Baltimore, MD, 2004), 373–419. Google Scholar

[Jac09] Jacquet, H., ‘Archimedean Rankin-Selberg integrals’, in Automorphic Forms and -functions II. Local Aspects (Contemp. Math.) vol. 489 (Amer. Math. Soc., Providence, RI, 2009), 57–172. Google Scholar

[Jac10] Jacquet, H., ‘Distinction by the quasi-split unitary group’, Israel J. Math. 178 (2010), 269–324. Google Scholar | DOI

[JLR99] Jacquet, H., Lapid, E. and Rogawski, J., ‘Periods of automorphic forms’, J. Amer. Math. Soc. 12(1) (1999), 173–240. Google Scholar | DOI

[JPSS79] Jacquet, H., Piatetski-Shapiro, I. and Shalika, J., ‘Automorphic forms on’, I. Ann. of Math. (2) 109(1) (1979), 169–212. Google Scholar | DOI

[JPSS83] Jacquet, H., Piatetskii-Shapiro, I. I. and Shalika, J. A., ‘Rankin-Selberg convolutions’, Amer. J. Math. 105(2) (1983), 367–464. Google Scholar | DOI

[JR11] Jacquet, H. and Rallis, S., ‘On the Gross-Prasad conjecture for unitary groups’, in On Certain -functions (Clay Math. Proc.) vol. 13 (Amer. Math. Soc., Providence, RI, 2011), 205–264. Google Scholar

[JS81a] Jacquet, H. and Shalika, J. A., ‘On Euler products and the classification of automorphic forms II’. Amer. J. Math. 103(4) (1981), 777–815. Google Scholar | DOI

[JS81b] Jacquet, H. and Shalika, J. A., ‘On Euler products and the classification of automorphic representations. I’, Amer. J. Math. 103(3) (1981), 499–558. Google Scholar | DOI

[JSZ10] Jiang, D., Sun, B. and Zhu, C.-B., ‘Uniqueness of Bessel models: the Archimedean case’, Geom. Funct. Anal. 20(3) (2010), 690–709. Google Scholar | DOI

[JZ20] Jiang, D. and Zhang, L., ‘Arthur parameters and cuspidal automorphic modules of classical groups’, Ann. of Math. (2) 191(3) (2020), 739–827. Google Scholar | DOI

[Kem15] Kemarsky, A., ‘A note on the Kirillov model for representations of ’, C. R. Math. Acad. Sci. Paris 353(7) (2015), 579–582. Google Scholar | DOI

[Kho08] Khoury, M. Jr., ‘Multiplicity-one results and explicit formulas for quasi-split p-adic unitary groups’, PhD thesis, The Ohio State University, ProQuest LLC, Ann Arbor, MI, 2008. Google Scholar

[KMSW] Kaletha, T, Minguez, A., Shin, S.-W. and White, P.-J., ‘Endoscopic classification of representations: Inner forms of unitary groups’, Preprint, 2014, . Google Scholar | arXiv

[Lap06] Lapid, E., ‘On the fine spectral expansion of Jacquet’s relative trace formula’, J. Inst. Math. Jussieu 5(2) (2006), 263–308. Google Scholar | DOI

[Lap13] Lapid, E., ‘On the Harish-Chandra Schwartz space of ’, in Automorphic Representations and -functions (Tata Inst. Fundam. Res. Stud. Math.) vol. 22 (Tata Inst. Fund. Res., Mumbai, 2013), 335–377. With an appendix by Farrell Brumley. Google Scholar

[Liu16] Liu, Y., ‘Refined global Gan-Gross-Prasad conjecture for Bessel periods’, J. Reine Angew. Math. 717 (2016), 133–194. Google Scholar | DOI

[LM15] Lapid, E. and Mao, Z., ‘A conjecture on Whittaker-Fourier coefficients of cusp forms’, J. Number Theory 146 (2015), 448–505. Google Scholar | DOI

[LR03] Lapid, E. and Rogawski, J., ‘Periods of Eisenstein series: the Galois case’, Duke Math. J. 120(1) (2003), 153–226. Google Scholar | DOI

[LW13] 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]. Google Scholar | DOI

[Mok15] Mok, C. P., ‘Endoscopic classification of representations of quasi-split unitary groups’, Mem. Amer. Math. Soc. (1108) (2015), vi+248. Google Scholar | DOI

[MW94] Mœglin, C. and Waldspurger, J.-L., Décomposition spectrale et séries d’Eisenstein (Progress in Mathematics) vol. 113 (Birkhäuser Verlag, Basel, 1994). Une paraphrase de l’Écriture. [A paraphrase of Scripture]. Google Scholar

[Mül98] Müller, W., ‘The trace class conjecture in the theory of automorphic forms. II’, Geom. Funct. Anal. 8(2) (1998), 315–355. Google Scholar | DOI

[Ram18] Ramakrishnan, D., ‘A Theorem on GL(n) à la Tchebotarev’, Preprint, 2018, . Google Scholar | arXiv

[Sha81] Shahidi, F., ‘On certain -functions’, Amer. J. Math. 103(2) (1981), 297–355. Google Scholar | DOI

[Shi76] Shintani, T., ‘On an explicit formula for class- “Whittaker functions” on over -adic fields’, Proc. Japan Acad. 52(4) (1976), 180–182. Google Scholar

[Tad09] Tadić, M., ‘ and ’, in Automorphic Forms and -functions II. Local Aspects (Contemp. Math.) vol. 489 (Amer. Math. Soc., Providence, RI, 2009), 285–313. Google Scholar

[Trè67] Trèves, F., Topological Vector Spaces, Distributions and Kernels (Academic Press, New York-London, 1967). Google Scholar

[Wal03] Waldspurger, J.-L., ‘La formule de Plancherel pour les groupes -adiques (d’après Harish-Chandra)’, J. Inst. Math. Jussieu 2(2) (2003), 235–333. Google Scholar | DOI

[Xue19] 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. Google Scholar | DOI

[Zel80] Zelevinsky, A., ‘Induced representations of reductive -adic groups. II. On irreducible representations of GL(n)’, Ann. Sci. École Norm. Sup. (4) 13(2) (1980), 165–210. Google Scholar | DOI

[Zha14a] Zhang, W., ‘Automorphic period and the central value of Rankin-Selberg -function’, J. Amer. Math. Soc. 27 (2014), 541–612. Google Scholar | DOI

[Zha14b] Zhang, W., ‘Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups’, Ann. of Math. (2) 180(3) (2014), 971–1049. Google Scholar | DOI

[Zha18] Zhang, L., ‘Whittaker-Shintani functions for general linear groups over -adic fields’, Sci. China Math. 61(1) (2018), 95–110. Google Scholar | DOI

[Zyd20] Zydor, M., ‘Les formules des traces relatives de Jacquet–Rallis grossières’, J. Reine Angew. Math. 762 (2020), 195–259. Google Scholar | DOI

Cité par Sources :