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Transactions of the American Mathematical Society

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

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Applications of Langlands’ functorial lift of odd orthogonal groups
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by Henry H. Kim PDF
Trans. Amer. Math. Soc. 354 (2002), 2775-2796 Request permission

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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Additional Information
  • Henry H. Kim
  • Affiliation: Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada
  • MR Author ID: 324906
  • Email: henrykim@math.toronto.edu
  • 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: https://doi.org/10.1090/S0002-9947-02-02969-0
  • MathSciNet review: 1895203