AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
Endoscopic Classification of representations of Quasi-Split Unitary Groups
About this Title
Chung Pang Mok, Morningside Center of Mathematics, Beijing, Department of Mathematics, Purdue University
Publication: Memoirs of the American Mathematical Society
Publication Year:
2015; Volume 235, Number 1108
ISBNs: 978-1-4704-1041-4 (print); 978-1-4704-2226-4 (online)
DOI: https://doi.org/10.1090/memo/1108
Published electronically: October 21, 2014
MSC: Primary 22E55, 22E50; Secondary 20G35, 11R42
Table of Contents
Chapters
- 1. Introduction
- 2. Statement of the main theorems
- 3. Local character identities and the intertwining relation
- 4. Trace formulas and their stabilization
- 5. The Standard model
- 6. Study of Critical Cases
- 7. Local Classification
- 8. Nontempered representations
- 9. Global classification
- 10. Addendum
Abstract
In this paper we establish the endoscopic classification of tempered representations of quasi-split unitary groups over local fields, and the endoscopic classification of the discrete automorphic spectrum of quasi-split unitary groups over global number fields. The method is analogous to the work of Arthur on orthogonal and symplectic groups, based on the theory of endoscopy and the comparison of trace formulas on unitary groups and general linear groups.- James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. MR 3135650
- James Arthur, The invariant trace formula. I. Local theory, J. Amer. Math. Soc. 1 (1988), no. 2, 323–383. MR 928262, DOI 10.1090/S0894-0347-1988-0928262-5
- James Arthur, The invariant trace formula. II. Global theory, J. Amer. Math. Soc. 1 (1988), no. 3, 501–554. MR 939691, DOI 10.1090/S0894-0347-1988-0939691-8
- James Arthur, A stable trace formula. I. General expansions, J. Inst. Math. Jussieu 1 (2002), no. 2, 175–277. MR 1954821, DOI 10.1017/S1474-748002000051
- James Arthur, A stable trace formula. II. Global descent, Invent. Math. 143 (2001), no. 1, 157–220. MR 1802795, DOI 10.1007/s002220000107
- James Arthur, A stable trace formula. III. Proof of the main theorems, Ann. of Math. (2) 158 (2003), no. 3, 769–873. MR 2031854, DOI 10.4007/annals.2003.158.769
- James Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84 (1989), no. 1, 19–84. MR 999488, DOI 10.1016/0022-1236(89)90110-9
- James Arthur, On a family of distributions obtained from Eisenstein series. II. Explicit formulas, Amer. J. Math. 104 (1982), no. 6, 1289–1336. MR 681738, DOI 10.2307/2374062
- James Arthur, Unipotent automorphic representations: global motivation, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 1–75. MR 1044818
- James Arthur, The $L^2$-Lefschetz numbers of Hecke operators, Invent. Math. 97 (1989), no. 2, 257–290. MR 1001841, DOI 10.1007/BF01389042
- James Arthur, On local character relations, Selecta Math. (N.S.) 2 (1996), no. 4, 501–579. MR 1443184, DOI 10.1007/PL00001383
- J.Arthur, A non-tempered intertwining relation for $\GL (N)$, in preparation.
- J.Arthur, Duality, Endoscopy and Hecke operators, in preparation.
- A. Aubert, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif $p$-adique. Trans. Amer. Math. Soc. 347 (1995), no. 6, 2179–2189. Erratum: Trans. Amer. Math. Soc. 348 (1996), no. 11, 4687–4690.
- Dubravka Ban, The Aubert involution and R-groups, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 5, 673–693 (English, with English and French summaries). MR 1951440, DOI 10.1016/S0012-9593(02)01105-9
- Nicolas Bergeron and Laurent Clozel, Spectre automorphe des variétés hyperboliques et applications topologiques, Astérisque 303 (2005), xx+218 (French, with English and French summaries). MR 2245761
- Joseph N. Bernstein, $P$-invariant distributions on $\textrm {GL}(N)$ and the classification of unitary representations of $\textrm {GL}(N)$ (non-Archimedean case), Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 50–102. MR 748505, DOI 10.1007/BFb0073145
- J. Bernstein, P. Deligne, and D. Kazhdan, Trace Paley-Wiener theorem for reductive $p$-adic groups, J. Analyse Math. 47 (1986), 180–192. MR 874050, DOI 10.1007/BF02792538
- A. Borel, Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 27–61. MR 546608
- W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
- W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207–231. MR 581582
- Pierre-Henri Chaudouard and Gérard Laumon, Le lemme fondamental pondéré. I. Constructions géométriques, Compos. Math. 146 (2010), no. 6, 1416–1506 (French, with English and French summaries). MR 2735371, DOI 10.1112/S0010437X10004756
- Pierre-Henri Chaudouard and Gérard Laumon, Le lemme fondamental pondéré. II. Énoncés cohomologiques, Ann. of Math. (2) 176 (2012), no. 3, 1647–1781 (French, with French summary). MR 2979859, DOI 10.4007/annals.2012.176.3.6
- Laurent Clozel and Patrick Delorme, Le théorème de Paley-Wiener invariant pour les groupes de Lie réductifs. II, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 2, 193–228 (French). MR 1046496
- Laurent Clozel, Michael Harris, and Jean-Pierre Labesse, Endoscopic transfer, On the stabilization of the trace formula, Stab. Trace Formula Shimura Var. Arith. Appl., vol. 1, Int. Press, Somerville, MA, 2011, pp. 475–496. MR 2856382
- J. W. Cogdell, I. I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the quasisplit classical groups, On certain $L$-functions, Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 117–140. MR 2767514
- Patrick Delorme and Paul Mezo, A twisted invariant Paley-Wiener theorem for real reductive groups, Duke Math. J. 144 (2008), no. 2, 341–380. MR 2437683, DOI 10.1215/00127094-2008-039
- A. Fröhlich and J. Queyrut, On the functional equation of the Artin $L$-function for characters of real representations, Invent. Math. 20 (1973), 125–138. MR 321888, DOI 10.1007/BF01404061
- Wee Teck Gan, Benedict H. Gross, and Dipendra Prasad, Symplectic local root numbers, central critical $L$ values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1–109 (English, with English and French summaries). Sur les conjectures de Gross et Prasad. I. MR 3202556
- D.Ginzburg, S.Rallis, S.Soudry, The descent map from automorphic representations of $\GL (n)$ to classical groups. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
- David Goldberg, Some results on reducibility for unitary groups and local Asai $L$-functions, J. Reine Angew. Math. 448 (1994), 65–95. MR 1266747, DOI 10.1515/crll.1994.448.65
- Harish-Chandra, Discrete series for semisimple Lie groups. II. Explicit determination of the characters, Acta Math. 116 (1966), 1–111. MR 219666, DOI 10.1007/BF02392813
- Harish-Chandra, Harmonic analysis on real reductive groups. I. The theory of the constant term, J. Functional Analysis 19 (1975), 104–204. MR 0399356, DOI 10.1016/0022-1236(75)90034-8
- Michael Harris and Jean-Pierre Labesse, Conditional base change for unitary groups, Asian J. Math. 8 (2004), no. 4, 653–683. MR 2127943
- Michael Harris and Richard Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich. MR 1876802
- Guy Henniart, Une preuve simple des conjectures de Langlands pour $\textrm {GL}(n)$ sur un corps $p$-adique, Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446, DOI 10.1007/s002220050012
- Guy Henniart, Correspondance de Langlands et fonctions $L$ des carrés extérieur et symétrique, Int. Math. Res. Not. IMRN 4 (2010), 633–673 (French). MR 2595008, DOI 10.1093/imrn/rnp150
- Kaoru Hiraga, On functoriality of Zelevinski involutions, Compos. Math. 140 (2004), no. 6, 1625–1656. MR 2098406, DOI 10.1112/S0010437X04000892
- H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
- H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms. II, Amer. J. Math. 103 (1981), no. 4, 777–815. MR 623137, DOI 10.2307/2374050
- R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), no. 1-4, 219–271. MR 909227, DOI 10.1007/BF01458070
- C. David Keys and Freydoon Shahidi, Artin $L$-functions and normalization of intertwining operators, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 1, 67–89. MR 944102
- Henry H. Kim and Muthukrishnan Krishnamurthy, Stable base change lift from unitary groups to $\textrm {GL}_n$, IMRP Int. Math. Res. Pap. 1 (2005), 1–52. MR 2149370
- Henry H. Kim and Muthukrishnan Krishnamurthy, Base change lift for odd unitary groups, Functional analysis VIII, Various Publ. Ser. (Aarhus), vol. 47, Aarhus Univ., Aarhus, 2004, pp. 116–125. MR 2127169
- Robert E. Kottwitz and Diana Shelstad, Foundations of twisted endoscopy, Astérisque 255 (1999), vi+190 (English, with English and French summaries). MR 1687096
- J.-P. Labesse, Changement de base CM et séries discrètes, On the stabilization of the trace formula, Stab. Trace Formula Shimura Var. Arith. Appl., vol. 1, Int. Press, Somerville, MA, 2011, pp. 429–470 (French). MR 2856380
- Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181
- Gérard Laumon and Bao Châu Ngô, Le lemme fondamental pour les groupes unitaires, Ann. of Math. (2) 168 (2008), no. 2, 477–573 (French, with English summary). MR 2434884, DOI 10.4007/annals.2008.168.477
- Jean-Pierre Labesse and Jean-Loup Waldspurger, La formule des traces tordue d’après le Friday Morning Seminar, CRM Monograph Series, vol. 31, American Mathematical Society, Providence, RI, 2013 (French). With a foreword by Robert Langlands [dual English/French text]. MR 3026269
- Paul Mezo, Character identities in the twisted endoscopy of real reductive groups, Mem. Amer. Math. Soc. 222 (2013), no. 1042, vi+94. MR 3076427, DOI 10.1090/S0065-9266-2012-00661-7
- Colette Mœglin, Classification et changement de base pour les séries discrètes des groupes unitaires $p$-adiques, Pacific J. Math. 233 (2007), no. 1, 159–204 (French, with English summary). MR 2366373, DOI 10.2140/pjm.2007.233.159
- Colette Mœglin and Marko Tadić, Construction of discrete series for classical $p$-adic groups, J. Amer. Math. Soc. 15 (2002), no. 3, 715–786. MR 1896238, DOI 10.1090/S0894-0347-02-00389-2
- C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752
- C. Mœglin and J.-L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics, vol. 113, Cambridge University Press, Cambridge, 1995. Une paraphrase de l’Écriture [A paraphrase of Scripture]. MR 1361168
- Sophie Morel, On the cohomology of certain noncompact Shimura varieties, Annals of Mathematics Studies, vol. 173, Princeton University Press, Princeton, NJ, 2010. With an appendix by Robert Kottwitz. MR 2567740
- Bao Châu Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169 (French). MR 2653248, DOI 10.1007/s10240-010-0026-7
- Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. MR 1081540
- J. D. Rogawski, Trace Paley-Wiener theorem in the twisted case, Trans. Amer. Math. Soc. 309 (1988), no. 1, 215–229. MR 957068, DOI 10.1090/S0002-9947-1988-0957068-2
- Peter Schneider and Ulrich Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 97–191. MR 1471867
- Freydoon Shahidi, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
- Freydoon Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), no. 4, 973–1007. MR 816396, DOI 10.1215/S0012-7094-85-05252-4
- Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
- D. Shelstad, $L$-indistinguishability for real groups, Math. Ann. 259 (1982), no. 3, 385–430. MR 661206, DOI 10.1007/BF01456950
- D. Shelstad, Tempered endoscopy for real groups. III. Inversion of transfer and $L$-packet structure, Represent. Theory 12 (2008), 369–402. MR 2448289, DOI 10.1090/S1088-4165-08-00337-3
- D. Shelstad, Tempered endoscopy for real groups. II. Spectral transfer factors, Automorphic forms and the Langlands program, Adv. Lect. Math. (ALM), vol. 9, Int. Press, Somerville, MA, 2010, pp. 236–276. MR 2581952
- D. Shelstad, On geometric transfer in real twisted endoscopy, Ann. of Math. (2) 176 (2012), no. 3, 1919–1985. MR 2979862, DOI 10.4007/annals.2012.176.3.9
- Sug Woo Shin, Galois representations arising from some compact Shimura varieties, Ann. of Math. (2) 173 (2011), no. 3, 1645–1741. MR 2800722, DOI 10.4007/annals.2011.173.3.9
- Laurent Clozel, Michael Harris, Jean-Pierre Labesse, and Bao-Châu Ngô (eds.), On the stabilization of the trace formula, Stabilization of the Trace Formula, Shimura Varieties, and Arithmetic Applications, vol. 1, International Press, Somerville, MA, 2011. MR 2742611
- Marko Tadić, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 335–382. MR 870688
- M. Tadić, On characters of irreducible unitary representations of general linear groups, Abh. Math. Sem. Univ. Hamburg 65 (1995), 341–363. MR 1359141, DOI 10.1007/BF02953339
- J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–26. MR 546607
- David A. Vogan Jr., The unitary dual of $\textrm {GL}(n)$ over an Archimedean field, Invent. Math. 83 (1986), no. 3, 449–505. MR 827363, DOI 10.1007/BF01394418
- J.-L. Waldspurger, Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), no. 2, 153–236 (French). MR 1440722, DOI 10.1023/A:1000103112268
- J.-L. Waldspurger, Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5 (2006), no. 3, 423–525 (French, with English and French summaries). MR 2241929, DOI 10.1017/S1474748006000041
- J.-L. Waldspurger, L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 194 (2008), no. 908, x+261 (French, with English summary). MR 2418405, DOI 10.1090/memo/0908
- J.-L. Waldspurger, À propos du lemme fondamental pondéré tordu, Math. Ann. 343 (2009), no. 1, 103–174 (French, with English summary). MR 2448443, DOI 10.1007/s00208-008-0267-7
- Jean-Loup Waldspurger, Endoscopie et changement de caractéristique: intégrales orbitales pondérées, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 5, 1753–1818 (French, with English and French summaries). MR 2573190
- J.-L.Waldspurger, La formule des traces locale tordue. Preprint (2012) arXiv:1205.1100
- J.-L.Waldspurger, Préparation à la stabilisation de la formule des traces tordue I: endoscopie tordue sur un corps local. Preprint (2013). Available at http://www.math.jussieu.fr/~waldspur/
- J.-L.Waldspurger, Préparation à la stabilisation de la formule des traces tordue II: intégrales orbitales et endoscopie sur un corps local non-archimédien. Preprint (2013). Available at http://www.math.jussieu.fr/~waldspur/
- Wei Zhang, Automorphic period and the central value of Rankin-Selberg L-function, J. Amer. Math. Soc. 27 (2014), no. 2, 541–612. MR 3164988, DOI 10.1090/S0894-0347-2014-00784-0