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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)

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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