Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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.


Applications of Langlands’ functorial lift of odd orthogonal groups
HTML articles powered by AMS MathViewer

by Henry H. Kim PDF
Trans. Amer. Math. Soc. 354 (2002), 2775-2796 Request permission


Together with Cogdell, Piatetski-Shapiro and Shahidi, we proved earlier the existence of a weak functorial lift of a generic cuspidal representation of $SO_{2n+1}$ to $GL_{2n}$. Recently, Ginzburg, Rallis and Soudry obtained a more precise form of the lift using their integral representation technique, namely, the lift is an isobaric sum of cuspidal representations of $GL_{n_i}$ (more precisely, cuspidal representations of $GL_{2n_i}$ such that the exterior square $L$-functions have a pole at $s=1$). One purpose of this paper is to give a simpler proof of this fact in the case that a cuspidal representation has one supercuspidal component. In a separate paper, we prove it without any condition using a result on spherical unitary dual due to Barbasch and Moy. We give several applications of the functorial lift: First, we parametrize square integrable representations with generic supercuspidal support, which have been classified by Moeglin and Tadic. Second, we give a criterion for cuspidal reducibility of supercuspidal representations of $GL_m\times SO_{2n+1}$. Third, we obtain a functorial lift from generic cuspidal representations of $SO_5$ to automorphic representations of $GL_5$, corresponding to the $L$-group homomorphism $Sp_4(\mathbb {C})\longrightarrow GL_5(\mathbb {C})$, given by the second fundamental weight.
  • Dan Barbasch and Allen Moy, Unitary spherical spectrum for $p$-adic classical groups, Acta Appl. Math. 44 (1996), no. 1-2, 3–37. Representations of Lie groups, Lie algebras and their quantum analogues. MR 1407038, DOI 10.1007/BF00116514
  • I. M. Sheffer, Some properties of polynomial sets of type zero, Duke Math. J. 5 (1939), 590–622. MR 81
  • William Casselman and Freydoon Shahidi, On irreducibility of standard modules for generic representations, Ann. Sci. École Norm. Sup. (4) 31 (1998), no. 4, 561–589 (English, with English and French summaries). MR 1634020, DOI 10.1016/S0012-9593(98)80107-9
  • J. Cogdell, H. Kim, I.I. Piatetski-Shapiro, and F. Shahidi, On lifting from classical groups to $GL_{N}$, Publ. Math. IHES 93 (2001), 5–30.
  • J. W. Cogdell and I. I. Piatetski-Shapiro, Converse theorems for $\textrm {GL}_n$, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 157–214. MR 1307299
  • David Ginzburg, Stephen Rallis, and David Soudry, Self-dual automorphic $\textrm {GL}_n$ modules and construction of a backward lifting from $\textrm {GL}_n$ to classical groups, Internat. Math. Res. Notices 14 (1997), 687–701. MR 1460389, DOI 10.1155/S1073792897000457
  • David Goldberg, $R$-groups and elliptic representations for similitude groups, Math. Ann. 307 (1997), no. 4, 569–588. MR 1464132, DOI 10.1007/s002080050051
  • M. Harris and R. Taylor, On the geometry and cohomology of some simple Shimura varieties, preprint, 1998.
  • 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
  • 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 representations. I, Amer. J. Math. 103 (1981), no. 3, 499–558. MR 618323, DOI 10.2307/2374103
  • Chris Jantzen, On supports of induced representations for symplectic and odd-orthogonal groups, Amer. J. Math. 119 (1997), no. 6, 1213–1262. MR 1481814
  • Chris Jantzen, On square-integrable representations of classical $p$-adic groups, Canad. J. Math. 52 (2000), no. 3, 539–581. MR 1758232, DOI 10.4153/CJM-2000-025-7
  • Henry H. Kim, Langlands-Shahidi method and poles of automorphic $L$-functions: application to exterior square $L$-functions, Canad. J. Math. 51 (1999), no. 4, 835–849. MR 1701344, DOI 10.4153/CJM-1999-036-0
  • Henry H. Kim, Langlands-Shahidi method and poles of automorphic $L$-functions. II, Israel J. Math. 117 (2000), 261–284. MR 1760595, DOI 10.1007/BF02773573
  • Henry H. Kim, Langlands-Shahidi method and poles of automorphic $L$-functions. II, Israel J. Math. 117 (2000), 261–284. MR 1760595, DOI 10.1007/BF02773573
  • —, Functoriality for exterior square of $GL_{4}$ and symmetric fourth of $GL_{2}$, submitted.
  • —, Residual spectrum of split classical groups; contribution from Borel subgroups, Pacific J. Math. 199 (2001), 417-445.
  • —, Residual spectrum of odd-orthogonal groups, IMRN 17 (2001), 873–906.
  • —, On local $L$-functions and normalized intertwining operators, preprint.
  • Stephen S. Kudla, The local Langlands correspondence: the non-Archimedean case, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 365–391. MR 1265559, DOI 10.2307/2118540
  • R. P. Langlands, Problems in the theory of automorphic forms, Lectures in modern analysis and applications, III, Lecture Notes in Math., Vol. 170, Springer, Berlin, 1970, pp. 18–61. MR 0302614
  • R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp. 101–170. MR 1011897, DOI 10.1090/surv/031/03
  • A. Borel and H. Jacquet, Automorphic forms and automorphic representations, 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. 189–207. With a supplement “On the notion of an automorphic representation” by R. P. Langlands. MR 546598
  • Jian-Shu Li, Some results on the unramified principal series of $p$-adic groups, Math. Ann. 292 (1992), no. 4, 747–761. MR 1157324, DOI 10.1007/BF01444646
  • Colette Mœglin, Représentations unipotentes et formes automorphes de carré intégrable, Forum Math. 6 (1994), no. 6, 651–744 (French, with English summary). MR 1300285, DOI 10.1515/form.1994.6.651
  • C. Mœglin, Orbites unipotentes et spectre discret non ramifié: le cas des groupes classiques déployés, Compositio Math. 77 (1991), no. 1, 1–54 (French). MR 1091891
  • —, Une conjecture sur le spectre résiduel des groupes classiques, preprint, 1994.
  • 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, DOI 10.1017/CBO9780511470905
  • 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. Moeglin and M. Tadic, Construction of discrete series for classical $p$-adic groups, preprint.
  • Goran Muić, Some results on square integrable representations; irreducibility of standard representations, Internat. Math. Res. Notices 14 (1998), 705–726. MR 1637097, DOI 10.1155/S1073792898000427
  • Dipendra Prasad and Dinakar Ramakrishnan, On the global root numbers of $\textrm {GL}(n)\times \textrm {GL}(m)$, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996) Proc. Sympos. Pure Math., vol. 66, Amer. Math. Soc., Providence, RI, 1999, pp. 311–330. MR 1703765
  • Dinakar Ramakrishnan, Modularity of the Rankin-Selberg $L$-series, and multiplicity one for $\textrm {SL}(2)$, Ann. of Math. (2) 152 (2000), no. 1, 45–111. MR 1792292, DOI 10.2307/2661379
  • 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
  • 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
  • Franz Rádl, Über die Teilbarkeitsbedingungen bei den gewöhnlichen Differential polynomen, Math. Z. 45 (1939), 429–446 (German). MR 82, DOI 10.1007/BF01580293
  • Freydoon Shahidi, On the Ramanujan conjecture and finiteness of poles for certain $L$-functions, Ann. of Math. (2) 127 (1988), no. 3, 547–584. MR 942520, DOI 10.2307/2007005
  • Freydoon Shahidi, Twisted endoscopy and reducibility of induced representations for $p$-adic groups, Duke Math. J. 66 (1992), no. 1, 1–41. MR 1159430, DOI 10.1215/S0012-7094-92-06601-4
  • Freydoon Shahidi, On multiplicativity of local factors, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 279–289. MR 1159120
  • 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
  • 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
  • Hiroyuki Yoshida, On the unitarizability of principal series representations of $\mathfrak {p}$-adic Chevalley groups, J. Math. Kyoto Univ. 32 (1992), no. 1, 155–233. MR 1145650, DOI 10.1215/kjm/1250519602
  • 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
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 22E55, 11F70
  • Retrieve articles in all journals with MSC (2000): 22E55, 11F70
Additional Information
  • Henry H. Kim
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada
  • MR Author ID: 324906
  • Email:
  • Received by editor(s): September 25, 2000
  • Received by editor(s) in revised form: February 21, 2001, and September 27, 2001
  • Published electronically: March 6, 2002
  • Additional Notes: Partially supported by NSF grant DMS9988672, NSF grant DMS9729992 (at IAS) and by Clay Mathematics Institute.
  • © Copyright 2002 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 2775-2796
  • MSC (2000): Primary 22E55, 11F70
  • DOI:
  • MathSciNet review: 1895203