Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

From Laplace to Langlands via representations of orthogonal groups


Authors: Benedict H. Gross and Mark Reeder
Journal: Bull. Amer. Math. Soc. 43 (2006), 163-205
MSC (2000): Primary 11S37, 20G05, 22E50
Published electronically: February 10, 2006
MathSciNet review: 2216109
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568–640. MR 0021942
  • 2. Hermann Boerner, Representation of groups with special consideration for the needs of modern physics, Translated from the German by P. G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr, North-Holland Publishing Co., Amsterdam; Interscience Publishers, a division of John Wiley & Sons, Inc., New York; 1963, 1963. MR 0148766
  • 3. Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629
  • 4. François Bruhat, 𝔭-adic groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 63–70. MR 0213356
  • 5. F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 0327923
  • 6. Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
  • 7. Bill Casselman, The 𝐿-group, Class field theory—its centenary and prospect (Tokyo, 1998) Adv. Stud. Pure Math., vol. 30, Math. Soc. Japan, Tokyo, 2001, pp. 217–258. MR 1846460
  • 8. Algebraic number theory, Proceedings of an instructional conference organized by the London Mathematical Society (a NATO Advanced Study Institute) with the support of the Inter national Mathematical Union. Edited by J. W. S. Cassels and A. Fröhlich, Academic Press, London; Thompson Book Co., Inc., Washington, D.C., 1967. MR 0215665
  • 9. S. DeBacker and M. Reeder, Depth-Zero Supercuspidal $ L$-packets and Their Stability, preprint, (2004), www2.bc.edu/~reederma/papers.html.
  • 10. P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 0393266
  • 11. Ferdinand Georg Frobenius, Gesammelte Abhandlungen. Bände I, II, III, Herausgegeben von J.-P. Serre, Springer-Verlag, Berlin-New York, 1968 (German). MR 0235974
  • 12. 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 0321888
  • 13. Benedict H. Gross, 𝐿-functions at the central critical point, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 527–535. MR 1265543
  • 14. Benedict H. Gross and Dipendra Prasad, On the decomposition of a representation of 𝑆𝑂_{𝑛} when restricted to 𝑆𝑂_{𝑛-1}, Canad. J. Math. 44 (1992), no. 5, 974–1002. MR 1186476, 10.4153/CJM-1992-060-8
  • 15. B. Gross and N. Wallach, Restriction of small discrete series representations to symmetric subgroups, The mathematical legacy of Harish-Chandra (Baltimore, MD, 1998) Proc. Sympos. Pure Math., vol. 68, Amer. Math. Soc., Providence, RI, 2000, pp. 255–272. MR 1767899, 10.1090/pspum/068/1767899
  • 16. T. Hagedorn, Multiplicities in Restricted Representations of $ GL_n({\mathbf{F}}_q)$, $ U_n({\mathbf{F}}_{q^2})$, $ SO_n({\mathbf{F}}_q)$, Ph.D. thesis, Harvard University, 1994.
  • 17. Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318. MR 0219665
  • 18. 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
  • 19. Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 0396855
  • 20. Guy Henniart, Une preuve simple des conjectures de Langlands pour 𝐺𝐿(𝑛) sur un corps 𝑝-adique, Invent. Math. 139 (2000), no. 2, 439–455 (French, with English summary). MR 1738446, 10.1007/s002220050012
  • 21. Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
  • 22. David Kazhdan and Yakov Varshavsky, Endoscopic decomposition of characters of certain cuspidal representations, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 11–20. MR 2048427, 10.1090/S1079-6762-04-00125-8
  • 23. Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, 10.1215/S0012-7094-84-05129-9
  • 24. R. Langlands, Digital Mathematics Archive, www.sunsite.ubc.ca/DigitalMathArchive/ Langlands.
  • 25. R. P. Langlands, Representations of abelian algebraic groups, Pacific J. Math. Special Issue (1997), 231–250. Olga Taussky-Todd: in memoriam. MR 1610871, 10.2140/pjm.1997.181.231
  • 26. George Lusztig, Classification of unipotent representations of simple 𝑝-adic groups, Internat. Math. Res. Notices 11 (1995), 517–589. MR 1369407, 10.1155/S1073792895000353
  • 27. G. Lusztig, Classification of unipotent representations of simple 𝑝-adic groups. II, Represent. Theory 6 (2002), 243–289. MR 1927955, 10.1090/S1088-4165-02-00173-5
  • 28. M. Reeder, On the restriction of Deligne-Lusztig characters, preprint (2005), www2.bc.edu/ ~reederma/papers.html.
  • 29. -, Some supercuspidal $ L$-packets of positive depth, preprint (2005), www2.bc.edu/ ~reederma/papers.html.
  • 30. Wilfried Schmid, Some properties of square-integrable representations of semisimple Lie groups, Ann. of Math. (2) 102 (1975), no. 3, 535–564. MR 0579165
  • 31. Wilfried Schmid, Discrete series, Representation theory and automorphic forms (Edinburgh, 1996) Proc. Sympos. Pure Math., vol. 61, Amer. Math. Soc., Providence, RI, 1997, pp. 83–113. MR 1476494, 10.1090/pspum/061/1476494
  • 32. J.-P. Serre, A course in arithmetic, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR 0344216
  • 33. Jean-Pierre Serre, Galois cohomology, Corrected reprint of the 1997 English edition, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. Translated from the French by Patrick Ion and revised by the author. MR 1867431
  • 34. Jean-Pierre Serre, Représentations linéaires des groupes finis, Third revised edition, Hermann, Paris, 1978 (French). MR 543841
  • 35. Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
  • 36. T. A. Springer, Reductive groups, 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. 3–27. MR 546587
  • 37. 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
  • 38. Elmar Thoma, Die Einschränkung der Charaktere von 𝐺𝐿(𝑛,𝑞) auf 𝐺𝐿(𝑛-1,𝑞), Math. Z. 119 (1971), 321–338 (German). MR 0288190
  • 39. J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, 1966, pp. 33–62. MR 0224710
  • 40. 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
  • 41. Jerrold B. Tunnell, Local 𝜀-factors and characters of 𝐺𝐿(2), Amer. J. Math. 105 (1983), no. 6, 1277–1307. MR 721997, 10.2307/2374441
  • 42. V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308
  • 43. Ernest B. Vinberg, Linear representations of groups, Basler Lehrbücher [Basel Textbooks], vol. 2, Birkhäuser Verlag, Basel, 1989. Translated from the Russian by A. Iacob. MR 1013789
  • 44. 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, 10.1090/conm/145/1216197
  • 45. Eugene P. Wigner, Group theory and its application to the quantum mechanics of atomic spectra, Expanded and improved ed. Translated from the German by J. J. Griffin. Pure and Applied Physics. Vol. 5, Academic Press, New York-London, 1959. MR 0106711

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2000): 11S37, 20G05, 22E50

Retrieve articles in all journals with MSC (2000): 11S37, 20G05, 22E50


Additional Information

Benedict H. Gross
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: gross@math.harvard.edu

Mark Reeder
Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
Email: reederma@bc.edu

DOI: http://dx.doi.org/10.1090/S0273-0979-06-01100-1
Received by editor(s): April 8, 2005
Published electronically: February 10, 2006
Additional Notes: The first author was supported by NSF grant DMS-0070674
The second author was supported by NSF grant DMS-0207231
Article copyright: © Copyright 2006 American Mathematical Society