Skip to Main Content

Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

From Laplace to Langlands via representations of orthogonal groups
HTML articles powered by AMS MathViewer

by Benedict H. Gross and Mark Reeder
Bull. Amer. Math. Soc. 43 (2006), 163-205
DOI: https://doi.org/10.1090/S0273-0979-06-01100-1
Published electronically: February 10, 2006
References
  • V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. (2) 48 (1947), 568–640. MR 21942, DOI 10.2307/1969129
  • Hermann Boerner, Representation of groups with special consideration for the needs of modern physics, North-Holland Publishing Co., Amsterdam; Interscience Publishers (a division of John Wiley & Sons, Inc.), New York, 1963. Translated from the German by P. G. Murphy in cooperation with J. Mayer-Kalkschmidt and P. Carr. MR 148766
  • 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, DOI 10.1007/978-3-540-89394-3
  • François Bruhat, ${\mathfrak {p}}$-adic groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, RI, 1966, pp. 63–70. MR 213356
  • F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923
  • 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
  • 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, DOI 10.2969/aspm/03010217
  • J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory, Academic Press, London; Thompson Book Co., Inc., Washington, DC, 1967. MR 215665
  • debacker-reeder:Lpackets S. DeBacker and M. Reeder, Depth-Zero Supercuspidal $L$-packets and Their Stability, preprint, (2004), www2.bc.edu/˜reederma/papers.html.
  • P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
  • Ferdinand Georg Frobenius, Gesammelte Abhandlungen. Bände I, II, III, Springer-Verlag, Berlin-New York, 1968 (German). Herausgegeben von J.-P. Serre. MR 235974
  • 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
  • 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, DOI 10.1090/pspum/055.1/1265543
  • Benedict H. Gross and Dipendra Prasad, On the decomposition of a representation of $\textrm {SO}_n$ when restricted to $\textrm {SO}_{n-1}$, Canad. J. Math. 44 (1992), no. 5, 974–1002. MR 1186476, DOI 10.4153/CJM-1992-060-8
  • 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, DOI 10.1090/pspum/068/1767899
  • hagedorn:thesis 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.
  • Harish-Chandra, Discrete series for semisimple Lie groups. I. Construction of invariant eigendistributions, Acta Math. 113 (1965), 241–318. MR 219665, DOI 10.1007/BF02391779
  • 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
  • Henryk Hecht and Wilfried Schmid, A proof of Blattner’s conjecture, Invent. Math. 31 (1975), no. 2, 129–154. MR 396855, DOI 10.1007/BF01404112
  • 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
  • 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
  • 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, DOI 10.1090/S1079-6762-04-00125-8
  • 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
  • langlands:archive R. Langlands, Digital Mathematics Archive, www.sunsite.ubc.ca/DigitalMathArchive/ Langlands.
  • R. P. Langlands, Representations of abelian algebraic groups, Pacific J. Math. Special Issue (1997), 231–250. Olga Taussky-Todd: in memoriam. MR 1610871, DOI 10.2140/pjm.1997.181.231
  • George Lusztig, Classification of unipotent representations of simple $p$-adic groups, Internat. Math. Res. Notices 11 (1995), 517–589. MR 1369407, DOI 10.1155/S1073792895000353
  • G. Lusztig, Classification of unipotent representations of simple $p$-adic groups. II, Represent. Theory 6 (2002), 243–289. MR 1927955, DOI 10.1090/S1088-4165-02-00173-5
  • reeder:dlrestriction M. Reeder, On the restriction of Deligne-Lusztig characters, preprint (2005), www2.bc.edu/ ˜reederma/papers.html. reeder:wild —, Some supercuspidal $L$-packets of positive depth, preprint (2005), www2.bc.edu/ ˜reederma/papers.html.
  • Wilfried Schmid, Some properties of square-integrable representations of semisimple Lie groups, Ann. of Math. (2) 102 (1975), no. 3, 535–564. MR 579165, DOI 10.2307/1971043
  • 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, DOI 10.1090/pspum/061/1476494
  • J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, No. 7, Springer-Verlag, New York-Heidelberg, 1973. Translated from the French. MR 344216
  • 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
  • Jean-Pierre Serre, Représentations linéaires des groupes finis, Third revised edition, Hermann, Paris, 1978 (French). MR 543841
  • 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
  • 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, RI, 1979, pp. 3–27. MR 546587
  • 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, RI, 1979, pp. 3–26. MR 546607
  • Elmar Thoma, Die Einschränkung der Charaktere von $\textrm {GL}(n,\,q)$ auf $\textrm {GL}(n-1,\,q)$, Math. Z. 119 (1971), 321–338 (German). MR 288190, DOI 10.1007/BF01109884
  • J. Tits, Classification of algebraic semisimple groups, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, RI, 1966, pp. 33–62. MR 224710
  • 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, RI, 1979, pp. 29–69. MR 546588
  • Jerrold B. Tunnell, Local $\epsilon$-factors and characters of $\textrm {GL}(2)$, Amer. J. Math. 105 (1983), no. 6, 1277–1307. MR 721997, DOI 10.2307/2374441
  • 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, DOI 10.1007/978-1-4612-1126-6
  • 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, DOI 10.1007/978-3-0348-0063-1
  • 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
  • Eugene P. Wigner, Group theory and its application to the quantum mechanics of atomic spectra, Pure and Applied Physics, Vol. 5, Academic Press, New York-London, 1959. Expanded and improved ed. Translated from the German by J. J. Griffin. MR 106711
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
Bibliographic Information
  • Benedict H. Gross
  • Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
  • MR Author ID: 77400
  • Email: gross@math.harvard.edu
  • Mark Reeder
  • Affiliation: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
  • Email: reederma@bc.edu
  • 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
  • © Copyright 2006 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 43 (2006), 163-205
  • MSC (2000): Primary 11S37, 20G05, 22E50
  • DOI: https://doi.org/10.1090/S0273-0979-06-01100-1
  • MathSciNet review: 2216109