Multiplicity one theorem for the GinzburgâRallis model: The tempered case
HTML articles powered by AMS MathViewer
- by Chen Wan PDF
- Trans. Amer. Math. Soc. 371 (2019), 7949-7994 Request permission
Abstract:
Following the method developed by Waldspurger and Beuzart-Plessis in their proof of the local GanâGrossâPrasad conjecture, we prove a local trace formula for the GinzburgâRallis model. By applying this trace formula, we prove the multiplicity one theorem for the GinzburgâRallis model over the tempered Vogan L-packets. In some cases, we also prove the epsilon dichotomy conjecture which gives a relation between the multiplicity and the exterior cube epsilon factor. This is a sequel to another work of ours in which we proved the geometric side of the trace formula.References
- Raphaël Beuzart-Plessis, La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes unitaires, Mém. Soc. Math. Fr. (N.S.) 149 (2016), vii+191 (French, with English and French summaries). MR 3676153, DOI 10.24033/msmf.457
- R. Beuzart-Plessis, A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: The archimedean case, arXiv:1506.01452 [math.RT] (2015).
- Joseph N. Bernstein, On the support of Plancherel measure, J. Geom. Phys. 5 (1988), no. 4, 663â710 (1989). MR 1075727, DOI 10.1016/0393-0440(88)90024-1
- J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive $p$-adic groups, J. Analyse Math. 47 (1986), 180â192. MR 874050, DOI 10.1007/BF02792538
- Joseph Bernstein and Bernhard Krötz, Smooth FrĂ©chet globalizations of Harish-Chandra modules, Israel J. Math. 199 (2014), no. 1, 45â111. MR 3219530, DOI 10.1007/s11856-013-0056-1
- P. Deligne, D. Kazhdan, and M.-F. VignĂ©ras, ReprĂ©sentations des algĂšbres centrales simples $p$-adiques, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 33â117 (French). MR 771672
- David Ginzburg and Stephen Rallis, The exterior cube $L$-function for $\textrm {GL}(6)$, Compositio Math. 123 (2000), no. 3, 243â272. MR 1795291, DOI 10.1023/A:1002461508749
- Harish-Chandra, Harmonic analysis on real reductive groups. III. The Maass-Selberg relations and the Plancherel formula, Ann. of Math. (2) 104 (1976), no. 1, 117â201. MR 439994, DOI 10.2307/1971058
- Dihua Jiang, Residues of Eisenstein series and related problems, Eisenstein series and applications, Progr. Math., vol. 258, BirkhĂ€user Boston, Boston, MA, 2008, pp. 187â204. MR 2402684, DOI 10.1007/978-0-8176-4639-4_{6}
- HervĂ© Jacquet and Joseph Shalika, Rankin-Selberg convolutions: Archimedean theory, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 125â207. MR 1159102
- Dihua Jiang, Binyong Sun, and Chen-Bo Zhu, Uniqueness of Ginzburg-Rallis models: the Archimedean case, Trans. Amer. Math. Soc. 363 (2011), no. 5, 2763â2802. MR 2763736, DOI 10.1090/S0002-9947-2010-05285-7
- Robert E. Kottwitz, Harmonic analysis on reductive $p$-adic groups and Lie algebras, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 393â522. MR 2192014
- Friedrich Knop, Bernhard Krötz, Eitan Sayag, and Henrik Schlichtkrull, Simple compactifications and polar decomposition of homogeneous real spherical spaces, Selecta Math. (N.S.) 21 (2015), no. 3, 1071â1097. MR 3366926, DOI 10.1007/s00029-014-0174-6
- Friedrich Knop, On the set of orbits for a Borel subgroup, Comment. Math. Helv. 70 (1995), no. 2, 285â309. MR 1324631, DOI 10.1007/BF02566009
- Hung Yean Loke, Trilinear forms of $\mathfrak {gl}_2$, Pacific J. Math. 197 (2001), no. 1, 119â144. MR 1810211, DOI 10.2140/pjm.2001.197.119
- Chufeng Nien, Models of representations of general linear groups over p-adic fields, ProQuest LLC, Ann Arbor, MI, 2006. Thesis (Ph.D.)âUniversity of Minnesota. MR 2709083
- Dipendra Prasad, Trilinear forms for representations of $\textrm {GL}(2)$ and local $\epsilon$-factors, Compositio Math. 75 (1990), no. 1, 1â46. MR 1059954
- F. Rodier, ModĂšle de Whittaker et caractĂšres de reprĂ©sentations, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974), Lecture Notes in Math., Vol. 466, Springer, Berlin, 1975, pp. 151â171 (French). MR 0393355
- Yiannis Sakellaridis and Akshay Venkatesh, Periods and harmonic analysis on spherical varieties, Astérisque 396 (2017), viii+360 (English, with English and French summaries). MR 3764130
- J.-L. Waldspurger, La formule de Plancherel pour les groupes $p$-adiques (dâaprĂšs Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235â333 (French, with French summary). MR 1989693, DOI 10.1017/S1474748003000082
- J.-L. Waldspurger, Une formule intĂ©grale reliĂ©e Ă la conjecture locale de Gross-Prasad, Compos. Math. 146 (2010), no. 5, 1180â1290 (French, with English summary). MR 2684300, DOI 10.1112/S0010437X10004744
- Jean-Loup Waldspurger, Une formule intĂ©grale reliĂ©e Ă la conjecture locale de Gross-Prasad, 2e partie: extension aux reprĂ©sentations tempĂ©rĂ©es, AstĂ©risque 346 (2012), 171â312 (French, with English and French summaries). Sur les conjectures de Gross et Prasad. I. MR 3202558
- Chen Wan, A local relative trace formula for the Ginzburg-Rallis model: The geometric side, arXiv:1608.03837 [math.RT] (2016). Memoirs of the AMS (to appear).
- Chen Wan, A Local Trace Formula and the Multiplicity One Theorem for the Ginzburg-Rallis Model, ProQuest LLC, Ann Arbor, MI, 2017. Thesis (Ph.D.)âUniversity of Minnesota. MR 3768394
Additional Information
- Chen Wan
- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
- MR Author ID: 1230762
- Email: chenwan@mit.edu
- Received by editor(s): January 2, 2018
- Received by editor(s) in revised form: May 13, 2018
- Published electronically: November 2, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7949-7994
- MSC (2010): Primary 22E35, 22E50
- DOI: https://doi.org/10.1090/tran/7690
- MathSciNet review: 3955540