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Applications of Langlands' functorial lift of odd orthogonal groups


Author: Henry H. Kim
Journal: Trans. Amer. Math. Soc. 354 (2002), 2775-2796
MSC (2000): Primary 22E55, 11F70
DOI: https://doi.org/10.1090/S0002-9947-02-02969-0
Published electronically: March 6, 2002
MathSciNet review: 1895203
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Abstract: 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.


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  • [B-Mo] D. Barbasch and A. Moy, Unitary spherical spectrum for $p$-adic classical groups, Acta Appl. Math. 44 (1996), 3-37. MR 98k:22067
  • [Bo] A. Borel, Automorphic L-functions, Proc. Sympos. Pure Math. 33, part 2 (1979), 27-61. MR 81m:11056
  • [Ca-Sh] W. Casselman and F. Shahidi, On irreducibility of standard modules for generic representations, Ann. Sci. École Norm. Sup. 31 (1998), 561-589. MR 99f:22028
  • [C-Ki-PS-S] 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.
  • [Co-PS] J. Cogdell and I.I. Piatetski-Shapiro, Converse theorems for $\operatorname{GL}_n$, Publ. Math. IHES 79 (1994), 157-214. MR 95m:22009
  • [G-R-S] D. Ginzburg, S. Rallis and D. Soudry, Self-dual automorphic $GL_{n}$ modules and construction of a backward lifting from $GL_{n}$ to classical groups, IMRN 14 (1997). MR 98e:22013
  • [Go] D. Goldberg, $R$-groups and elliptic representations for similitude groups, Math. Ann. 307 (1997), 569-588. MR 98i:22024
  • [H-T] M. Harris and R. Taylor, On the geometry and cohomology of some simple Shimura varieties, preprint, 1998.
  • [He] G. Henniart, Une preuve simple des conjectures de Langlands pour $GL(n)$ sur un corps $p$-adique, Inv. Math. 139 (2000), 439-455. MR 2001e:11052
  • [J-PS-S] H. Jacquet, I.I. Piatetski-Shapiro and J. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), 367-464. MR 85g:11044
  • [J-S] -, On Euler products and the classification of automorphic forms I,II, Amer. J. Math. 103 (1981), 499-558; 777-815. MR 82m:10050b
  • [Ja1] C. Jantzen, On supports of induced representations for symplectic and odd-orthogonal groups, Amer. J. Math. 119 (1997), 1213-1262. MR 99b:22028
  • [Ja2] -, On square-integrable representations of classical $p$-adic groups, Can. J. Math. 52 (2000), 539-581. MR 2001f:22056
  • [Ki1] H. Kim, Langlands-Shahidi method and poles of automorphic $L$-functions: application to exterior square $L$-functions, Can. J. Math. 51 (1999), 835-849. MR 2000f:11058
  • [Ki2] -, Langlands-Shahidi method and poles of automorphic $L$-functions II, Israel J. Math. 117 (2000), 261-284. MR 2001i:11059a
  • [Ki3] -, Correction to: Langlands-Shahidi method and poles of automorphic $L$-functions II, Israel J. Math. 118 (2000), 379. MR 2001i:11059b
  • [Ki4] -, Functoriality for exterior square of $GL_{4}$ and symmetric fourth of $GL_{2}$, submitted.
  • [Ki5] -, Residual spectrum of split classical groups; contribution from Borel subgroups, Pacific J. Math. 199 (2001), 417-445.
  • [Ki6] -, Residual spectrum of odd-orthogonal groups, IMRN 17 (2001), 873-906. CMP 2002:02
  • [Ki7] -, On local $L$-functions and normalized intertwining operators, preprint.
  • [Ku] S. Kudla, The local Langlands correspondence: the non-archimedean case, in Proceedings of Symposia in Pure Mathematics, vol. 55, part 2, American Mathematical Society, 1994, pp. 365-391. MR 95d:11065
  • [La1] R.P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., vol. 170, Springer-Verlag, Berlin-Heidelberg-New York, 1970, pp. 18-86. MR 46:1758
  • [La2] -, On the classification of irreducible representations of real algebraic groups, in Representation Theory and Harmonic Analysis on Semisimple Lie groups (P.J. Sally, Jr. and D.A. Vogan, ed.), Mathematical Surveys and Monographs, vol. 31, AMS, 1989, pp. 101-170. MR 91e:22017
  • [La3] -, On the notion of an automorphic representation, Proc. Symp. Pure Math. 33, part 1, 1979, pp. 189-207. MR 81m:10055
  • [Li] J-S. Li, Some results on unramified principal series of $p$-adic groups, Math. Ann. 292 (1992), 747-761. MR 93d:22023
  • [M1] C. Moeglin, Représentations unipotentes et formes automorphes de carré intégrable, Forum mathematicum 6 (1994), 651-744. MR 95k:22024
  • [M2] -, Orbites unipotentes et spectre discret non ramifie, Le cas des groupes classiques déploys, Comp. Math. 77 (1991), 1-54. MR 92d:11054
  • [M3] -, Une conjecture sur le spectre résiduel des groupes classiques, preprint, 1994.
  • [M-W1] C. Moeglin and J.L. Waldspurger, Spectral Decomposition and Eisenstein series, une paraphrase de l'Ecriture, Cambridge tracts in mathematics, vol 113, Cambridge University Press, 1995. MR 97d:11083
  • [M-W2] C. Moeglin and J.L. Waldspurger, Le spectre résiduel de $GL(n)$, Ann. Scient. Éc. Norm. Sup. 22 (1989), 605-674. MR 91b:22028
  • [M-Ta] C. Moeglin and M. Tadic, Construction of discrete series for classical $p$-adic groups, preprint.
  • [Mu] G. Muic, Some results on square integrable representations; irreducibility of standard representations, IMRN 14 (1998), 705-726. MR 99f:22031
  • [P-R] D. Prasad and D. Ramakrishnan, On the global root numbers of $GL(n)\times GL(m)$, Proc. Symp. in Pure Math. 66, 1999, pp. 311-330. MR 2000f:11060
  • [Ra] D. Ramakrishnan, Modularity of the Rankin-Selberg $L$-series, and multiplicity one for $SL(2)$, Ann. of Math. 152 (2000), 45-111. MR 2001g:11077
  • [Ro] J. Rogawski, Representations of $GL(n)$ and division algebras over a $p$-adic field, Duke Math. J. 50 (1983), 161-196. MR 84j:12018
  • [Sh1] F. Shahidi, A proof of Langlands conjecture on Plancherel measures; complementary series for $p$-adic groups, Annals of Math. 132 (1990), 273-330. MR 91m:11095
  • [Sh2] -, On certain $L$-functions, Amer. J. Math 103 (1981), 297-355. MR 82i:11030
  • [Sh3] -, On the Ramanujan conjecture and finiteness of poles for certain $L$-functions, Ann. of Math. 127 (1988), 547-584. MR 89h:11021
  • [Sh4] -, Twisted endoscopy and reducibility of induced representations for $p$-adic groups, Duke Math. J. 66, No. 1 (1992), 1-41. MR 93b:22034
  • [Sh5] -, On multiplicativity of local factors, In: Festschrift in honor of I.I. Piatetski-Shapiro, Part II, Israel Math. Conf. Proc. 3, Weizmann, Jerusalem, 1990, pp. 279-289. MR 93e:11144
  • [Sh6] -, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 973-1007. MR 87m:11049
  • [Ta] M. Tadic, Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case), Ann. Sci. Ec. Norm. Sup. 19 (1986), 335-382. MR 88b:22021
  • [Yo] H. Yoshida, On the unitarizability of principal series representations of $p$-adic Chevalley groups, J. Math. Kyoto Univ. 32, No 1 (1992), 155-233. MR 93c:22035
  • [Ze] A. Zelevinsky, Induced representations of reductive $p$-adic groups. II, Ann. Sci. Ec. Norm. Sup. 13 (1980), 165-210. MR 83g:22012

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Additional Information

Henry H. Kim
Affiliation: Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada
Email: henrykim@math.toronto.edu

DOI: https://doi.org/10.1090/S0002-9947-02-02969-0
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.
Article copyright: © Copyright 2002 American Mathematical Society

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