Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-02T21:08:52.660Z Has data issue: false hasContentIssue false

Spherical Fundamental Lemma for Metaplectic Groups

Published online by Cambridge University Press:  20 November 2018

Caihua Luo*
Affiliation:
Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076 email: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we prove the spherical fundamental lemma for metaplectic group $M{{p}_{2n}}$ based on the formalism of endoscopy theory by J. Adams, D. Renard, and W.-W. Li.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

References

[Art78] Arthur, J. G., A trace formula for reductive groups. I. Terms associated to classes in G(ℚ). Duke Math. J. 45(1978), no. 4, 911952. http://dx.doi.org/10.1215/S0012-7094-78-04542-8 Google Scholar
[Art88] Arthur, J. G., The invariant trace formula. II. Global theory. J. Amer. Math. Soc. 1(1988), no. 3, 501554. http://dx.doi.org/10.1090/S0894-0347-1988-0939691-8 Google Scholar
[ArtO5] Arthur, J. G., An introduction to the trace formula. In: Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., 4, American Mathematical Society, Providence, RI, 2005, pp. 1263.Google Scholar
[CD84] Clozel, L. and Delorme, P., Le théoreme de Paley-Wiener invariant pour les groupes de Lie réductifs. Invent. Math. 77(1984), no. 3, 427453. http://dx.doi.Org/10.1007/BF01388832 Google Scholar
[Clo85] Clozel, L., Sur une conjecture de Howe. I. Compositio Math. 56(1985), no. 1, 87110.Google Scholar
[Clo89] Clozel, L., Orbital integrals on p-adic groups: a proof of the Howe conjecture. Ann. of Math. 129(1989), no. 2, 237251. http://dx.doi.Org/10.2307/1971447 Google Scholar
[Clo90] Clozel, L., The fundamental lemma for stable base change. Duke Math. J. 61(1990), no. 1, 255302. http://dx.doi.org/10.1215/S0012-7094-90-06112-5 Google Scholar
[DG] Demazure, M. and Grothendieck, Alexandre, Schémas en groupes, Tome III: Structure des schémas en groupes réductifs. Lecture Notes in Mathematics, 153, Springer, New York, 1970, pp. 19621964.Google Scholar
[Duf75] Duflo, M., Représentations irréductibles des groupes semi-simples complexes. In: Analyse harmonique sur les groupes de Lie (Sém., Nancy-Strasbourg, 1973-75), Lectures Notes in Math., 497, Springer, Berlin, 1975, pp. 2688.Google Scholar
[GGar] Gan, W. T. and Gao, F., The Langlands-Weissman program for Brylinski-Deligne extensions. Asterisque, to appear.Google Scholar
[GS12] Gan, W. T. and Savin, G., Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence. Compos. Math. 148(2012), no. 06, 16551694. http://dx.doi.Org/10.1112/S0010437X12000486 Google Scholar
[Hal95] Hales, T. C., On the fundamental lemma for standard endoscopy: reduction to unit elements. Canad. J. Math. 47(1995), no. 5, 974994. http://dx.doi.org/10.4153/CJM-1995-051-5 Google Scholar
[HC59] Harish-Chandra, , Automorphic forms on a semisimple Lie group. Proc. Nat. Acad. Sci. U. S. A. 45(1959), 570573. http://dx.doi.Org/10.1073/pnas.45.4.570 Google Scholar
[Key82] Keys, D., Reducibility of unramified unitary principal series representations of p-adic groups and class-1 representations. Math. Ann. 260(1982), no. 4, 397402. http://dx.doi.org/10.1007/BF01457019 Google Scholar
[Kot86] Kottwitz, R. E., Stable trace formula: elliptic singular terms. Math. Ann. 275(1986), no. 3, 365399. http://dx.doi.org/10.1007/BF01458611 Google Scholar
[KR00] Kottwitz, R. E. and Rogawski, J. D., The distributions in the invariant trace formula are supported on characters. Canad. J. Math. 52(2000), no. 4, 804814. http://dx.doi.org/10.4153/CJM-2000-034-6 Google Scholar
[Kud96] Kudla, S., Notes on the local theta correspondence. 1996. http://www.math.toronto.edu/skudla/castle.pdf Google Scholar
[Lab0l] Labesse, J.-P., Nombres de Tamagawa des groupes réductifs quasi-connexes. Manuscripta Math. 104(2001), no. 4, 407430. http://dx.doi.org/10.1007/s002290170016 Google Scholar
[Lill] Li, W.-W., Transfert des intégrates orbitales pour legroupe métaplectique. Compos. Math. 147(2011), no. 2, 524590. http://dx.doi.Org/10.1112/S0010437X10004963 Google Scholar
[Li12a] Li, W.-W., La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale. Ann. Sci. Ec. Norm. Super. 45(2012), no. 5, 787859. http://dx.doi.org/10.24033/asens.2178 Google Scholar
[Li12b] Li, W.-W., Le lemme fondamental pondéré pour le groupe métaplectique. Canad. J. Math. 64(2012), no. 3, 497543. http://dx.doi.org/10.4153/CJM-2O11-088-1 Google Scholar
[Li14a] Li, W.-W., La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin. J. Reine Angew. Math. 686(2014), 37109. http://dx.doi.org/10.1515/crelle-2012-0015 Google Scholar
[Li14b] Li, W.-W., La formule des traces pour les revêtements de groupes réductifs connexes. IV. Distributions invariantes. Ann. Inst. Fourier (Grenoble) 64(2014), no. 6, 23792448. http://dx.doi.org/10.5802/aif.2915 Google Scholar
[Li15] Li, W.-W., Laformule des traces stable pour legroupe métaplectique: les termes elliptiques. Invent. Math. 202(2015), no. 2, 743838. http://dx.doi.Org/10.1007/s00222-015-0577-9 Google Scholar
[Li16] Li, W.-W., Spectral transfer for metaplectic groups. I. Local character relations. J. Inst. Math. Jussieu, 1-99. http://dx.doi.Org/10.1017/S1474748016000384 Google Scholar
[LMW15] Lemaire, B., Mœglin, C., and Waldspurger, J.-L., Le lemme fondamental pour l'endoscopie tordue: réduction aux éléments unités. arxiv:1506.03383Google Scholar
[LW15] Lemaire, B. and Waldspurger, J.-L., Le lemme fondamental pour l'endoscopie tordue: le cas où legroupe endoscopique non ramifié est un tore. arxiv:1511.08606Google Scholar
[LP81] Lion, G. and Perrin, P., Extension des representations de groupes unipotents p-adiques Calculs d'obstructions. In: Noncommutative harmonic analysis and Lie groups (Marseille, 1980), Lecture Notes in Math., 880, Springer, Berlin-New York, 1981, pp. 337356.Google Scholar
[Luoar] Luo, C., Howe finiteness conjecture for covering groups. Math. Res. Lett., to appear.Google Scholar
[Si179] Silberger, A. J., Introduction to harmonic analysis on reductive p-adic groups. Based on lectures by Harish-Chandra at The Institute for Advanced Study, 1971-73. Mathematical Notes, 23, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1979.Google Scholar
[Szp13] Szpruch, D., Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method. Israel J. Math. 195(2013), no. 2, 897971. http://dx.doi.Org/10.1007/s11856-012-0140-y Google Scholar
[Tad83] Tadić, M., Harmonic analysis of spherical functions on reductive groups over p-adic fields. Pacific J. Math. 109(1983), no. 1, 215235. http://dx.doi.Org/10.2140/pjm.1983.109.215 Google Scholar
[Vig82] Vignéras, M.-F., Caractérisation des intégrates orbitales sur un groupe réductif p-adique. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(1982), 945961.Google Scholar
[Wa103] Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques. D'apres Harish-Chandra. J. Inst. Math. Jussieu 2(2003), no. 2, 235333. http://dx.doi.Org/10.1017/S1474748003000082 Google Scholar
[Wei14] Weissman, M. H., Covers of tori over local and global fields. Amer. J. Math. 138(2016), no. 6, 15331573. http://dx.doi.org/10.1353/ajm.2016.0046 Google Scholar
[Wei15] Weissman, M. H., L-groups and parameters for covering groups. arxiv:1507.01042Google Scholar