AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution
On $L$-packets for inner forms of $SL_n$
About this Title
Kaoru Hiraga, Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan and Hiroshi Saito, Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2012; Volume 215, Number 1013
ISBNs: 978-0-8218-5364-1 (print); 978-0-8218-8519-2 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00642-8
Published electronically: April 21, 2011
MSC: Primary 22E50; Secondary 11F70, 22E55
Table of Contents
Chapters
- 1. Introduction
- 2. Restriction of Representations
- 3. Whittaker Normalization over Local Fields
- 4. Restriction of Cusp Forms
- 5. Whittaker Normalization over Global Fields
- 6. Endoscopy and Its Automorphisms
- 7. A Conjectural Formula for Endoscopic Transfer
- 8. Descent to Levi Subgroups
- 9. Relevance Conditions for Langlands Parameters
- 10. Endoscopy for Inner Forms of $GL_n$
- 11. Local Langlands Correspondence for Inner Forms of $GL_n$
- 12. $L$-packets for Inner Forms of $SL_n$
- 13. $L$-packets for Inner Forms of $SL_n$ over Archimedean Fields
- 14. Multiplicity Formula for $SL_n$
- 15. Multiplicity Formula for Inner Forms of $SL_n$
- 16. Lemmas for Trace Formula
- 17. Trace Formula
- A. Transfer Factors
Abstract
The theory of $L$-indistinguishability for inner forms of $SL_2$ has been established in the well-known paper of Labesse and Langlands ($L$-indistinguishability for $\textrm {SL}(2)$. Canad. J. Math. 31 (1979), no. 4, 726–785). In this paper, we study $L$-indistinguishability for inner forms of $SL_n$ for general $n$. Following the idea of Vogan in (The local Langlands conjecture. Representation theory of groups and algebras, 305–379, Contemp. Math. 145 (1993)), we modify the $S$-group and show that such an $S$-group fits well in the theory of endoscopy for inner forms of $SL_n$.- 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, Unipotent automorphic representations: conjectures, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
- 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, On elliptic tempered characters, Acta Math. 171 (1993), no. 1, 73–138. MR 1237898, DOI 10.1007/BF02392767
- James Arthur, On local character relations, Selecta Math. (N.S.) 2 (1996), no. 4, 501–579. MR 1443184, DOI 10.1007/PL00001383
- James Arthur, On the transfer of distributions: weighted orbital integrals, Duke Math. J. 99 (1999), no. 2, 209–283. MR 1708030, DOI 10.1215/S0012-7094-99-09909-X
- James Arthur, A note on $L$-packets, Pure Appl. Math. Q. 2 (2006), no. 1, Special Issue: In honor of John H. Coates., 199–217. MR 2217572, DOI 10.4310/PAMQ.2006.v2.n1.a9
- James Arthur and Laurent Clozel, Simple algebras, base change, and the advanced theory of the trace formula, Annals of Mathematics Studies, vol. 120, Princeton University Press, Princeton, NJ, 1989. MR 1007299
- Alexandru Ioan Badulescu, Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math. 172 (2008), no. 2, 383–438. With an appendix by Neven Grbac. MR 2390289, DOI 10.1007/s00222-007-0104-8
- 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
- Don Blasius, On multiplicities for $\textrm {SL}(n)$, Israel J. Math. 88 (1994), no. 1-3, 237–251. MR 1303497, DOI 10.1007/BF02937513
- 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
- P. Deligne, D. Kazhdan, and M.-F. Vignéras, Représentations des algèbres centrales simples $p$-adiques, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 33–117 (French). MR 771672
- Daniel Flath, A comparison of the automorphic representations of $\textrm {GL}(3)$ and its twisted forms, Pacific J. Math. 97 (1981), no. 2, 373–402. MR 641166
- S. S. Gelbart and A. W. Knapp, $L$-indistinguishability and $R$ groups for the special linear group, Adv. in Math. 43 (1982), no. 2, 101–121. MR 644669, DOI 10.1016/0001-8708(82)90030-5
- Thomas C. Hales, Unipotent representations and unipotent classes in $\textrm {SL}(n)$, Amer. J. Math. 115 (1993), no. 6, 1347–1383. MR 1254737, DOI 10.2307/2374969
- Thomas C. Hales, A simple definition of transfer factors for unramified groups, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 109–134. MR 1216184, DOI 10.1090/conm/145/1216184
- Thomas C. Hales, On the fundamental lemma for standard endoscopy: reduction to unit elements, Canad. J. Math. 47 (1995), no. 5, 974–994. MR 1350645, DOI 10.4153/CJM-1995-051-5
- Harish-Chandra, Harmonic analysis on reductive $p$-adic groups, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin-New York, 1970. Notes by G. van Dijk. MR 0414797
- Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, Lie theories and their applications (Proc. Ann. Sem. Canad. Math. Congr., Queen’s Univ., Kingston, Ont., 1977) Queen’s Univ., Kingston, Ont., 1978, pp. 281–347. Queen’s Papers in Pure Appl. Math., No. 48. MR 0579175
- 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, Représentations du groupe de Weil d’un corps local, Enseign. Math. (2) 26 (1980), no. 1-2, 155–172 (French). MR 590513
- 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, Représentations des groupes réductifs $p$-adiques et de leurs sous-groupes distingués cocompacts, J. Algebra 236 (2001), no. 1, 236–245 (French, with English and French summaries). MR 1808353, DOI 10.1006/jabr.2000.8497
- Guy Henniart and Rebecca Herb, Automorphic induction for $\textrm {GL}(n)$ (over local non-Archimedean fields), Duke Math. J. 78 (1995), no. 1, 131–192. MR 1328755, DOI 10.1215/S0012-7094-95-07807-7
- Rebecca A. Herb, Matching theorems for twisted orbital integrals, Pacific J. Math. 171 (1995), no. 2, 409–428. MR 1372236
- Kaoru Hiraga and Hiroshi Saito, On restriction of admissible representations, Algebra and number theory, Hindustan Book Agency, Delhi, 2005, pp. 299–326. MR 2193361
- H. Jacquet and R. P. Langlands, Automorphic forms on $\textrm {GL}(2)$, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR 0401654
- 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
- Helmut Koch, On the local Langlands conjecture for central division algebras of index $p$, Invent. Math. 62 (1980/81), no. 2, 243–268. MR 595588, DOI 10.1007/BF01389160
- Robert E. Kottwitz, Rational conjugacy classes in reductive groups, Duke Math. J. 49 (1982), no. 4, 785–806. MR 683003
- Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, DOI 10.1215/S0012-7094-84-05129-9
- Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365–399. MR 858284, DOI 10.1007/BF01458611
- Robert E. Kottwitz, Tamagawa numbers, Ann. of Math. (2) 127 (1988), no. 3, 629–646. MR 942522, DOI 10.2307/2007007
- 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, Cohomologie, $L$-groupes et fonctorialité, Compositio Math. 55 (1985), no. 2, 163–184 (French). MR 795713
- J.-P. Labesse and R. P. Langlands, $L$-indistinguishability for $\textrm {SL}(2)$, Canadian J. Math. 31 (1979), no. 4, 726–785. MR 540902, DOI 10.4153/CJM-1979-070-3
- J.-P. Labesse and J. Schwermer, On liftings and cusp cohomology of arithmetic groups, Invent. Math. 83 (1986), no. 2, 383–401. MR 818358, DOI 10.1007/BF01388968
- R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Springer, Berlin, 1970, pp. 18–61. Lecture Notes in Math., Vol. 170. MR 0302614
- R. P. Langlands, Stable conjugacy: definitions and lemmas, Canadian J. Math. 31 (1979), no. 4, 700–725. MR 540901, DOI 10.4153/CJM-1979-069-2
- —, Les débuts d’une formule des traces stable. Publications Mathématiques de l’Université Paris VII , 13. 1983.
- 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
- R. Langlands and D. Shelstad, Descent for transfer factors, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 485–563. MR 1106907
- C. Moeglin, Representations of $\textrm {GL}(n)$ over the real field, Representation theory and automorphic forms (Edinburgh, 1996) Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 157–166. MR 1476497, DOI 10.1090/pspum/061/1476497
- Bao Châu Ngô, Le lemme fondamental pour les algèbres de Lie. Preprint.
- Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory, Pure and Applied Mathematics, vol. 139, Academic Press, Inc., Boston, MA, 1994. Translated from the 1991 Russian original by Rachel Rowen. MR 1278263
- Jonathan D. Rogawski, Representations of $\textrm {GL}(n)$ and division algebras over a $p$-adic field, Duke Math. J. 50 (1983), no. 1, 161–196. MR 700135
- Jean-Pierre Serre, Cohomologie galoisienne, Lecture Notes in Mathematics, No. 5, Springer-Verlag, Berlin-New York, 1965 (French). With a contribution by Jean-Louis Verdier; Troisième édition, 1965. MR 0201444
- Freydoon Shahidi, Some results on $L$-indistinguishability for $\textrm {SL}(r)$, Canad. J. Math. 35 (1983), no. 6, 1075–1109. MR 738845, DOI 10.4153/CJM-1983-060-9
- J. A. Shalika, The multiplicity one theorem for $\textrm {GL}_{n}$, Ann. of Math. (2) 100 (1974), 171–193. MR 348047, DOI 10.2307/1971071
- D. Shelstad, $L$-indistinguishability for real groups, Math. Ann. 259 (1982), no. 3, 385–430. MR 661206, DOI 10.1007/BF01456950
- Diana Shelstad, Orbital integrals, endoscopic groups and $L$-indistinguishability for real groups, Conference on automorphic theory (Dijon, 1981) Publ. Math. Univ. Paris VII, vol. 15, Univ. Paris VII, Paris, 1983, pp. 135–219. MR 723184
- B. Speh, Some results on principal series of $\mathrm {GL}(n, R)$, Ph.D. Thesis, Mass. Inst. Tech., Cambridge, MA, 1977
- Birgit Speh and David Vogan, A reducibility criterion for generalized principal series, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 12, 5252. MR 457634, DOI 10.1073/pnas.74.12.5252
- Birgit Speh and David A. Vogan Jr., Reducibility of generalized principal series representations, Acta Math. 145 (1980), no. 3-4, 227–299. MR 590291, DOI 10.1007/BF02414191
- Marko Tadić, Induced representations of $\textrm {GL}(n,A)$ for $p$-adic division algebras $A$, J. Reine Angew. Math. 405 (1990), 48–77. MR 1040995, DOI 10.1515/crll.1990.405.48
- J. Tits, Reductive groups over local fields, 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. 29–69. MR 546588
- David A. Vogan Jr., Gel′fand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), no. 1, 75–98. MR 506503, DOI 10.1007/BF01390063
- David A. Vogan Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379. MR 1216197, DOI 10.1090/conm/145/1216197
- J.-L. Waldspurger, Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondamental, Canad. J. Math. 43 (1991), no. 4, 852–896 (French). MR 1127034, DOI 10.4153/CJM-1991-049-5
- 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
- Nolan R. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1024, Springer, Berlin, 1983, pp. 287–369. MR 727854, DOI 10.1007/BFb0071436
- André Weil, Exercices dyadiques, Invent. Math. 27 (1974), 1–22 (French). MR 379445, DOI 10.1007/BF01389962
- A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. II. On irreducible representations of $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084