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Some local-global non-vanishing results for theta lifts from orthogonal groups

Author: Shuichiro Takeda
Journal: Trans. Amer. Math. Soc. 361 (2009), 5575-5599
MSC (2000): Primary 11F27
Published electronically: April 10, 2009
MathSciNet review: 2515824
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Abstract: We, first, improve a theorem of B. Roberts which characterizes non-vanishing of a global theta lift from $ \operatorname{O}(X)$ to $ \operatorname{Sp}(n)$ in terms of non-vanishing of local theta lifts. In particular, we will remove all the Archimedean conditions imposed upon his theorem. Secondly, following Roberts, we will apply our theorem to theta lifting of low rank similitude groups. Namely we characterize the non-vanishing condition of a global theta lift from $ \operatorname{GO}(4)$ to $ \operatorname{GSp}(2)$ in our improved setting. Also we consider non-vanishing conditions of a global theta lift from $ \operatorname{GO}(4)$ to $ \operatorname{GSp}(1)$ and explicitly compute the lift when it exists.

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  • [A] J. Adams and D. Barbasch, Reductive dual pair correspondence for complex groups, J. Funct. Anal. 132 (1995), 1-42. MR 1346217 (96h:22003)
  • [AP] J. Adler and D. Prasad, On certain multiplicity one theorems, Israel J. Math. 153 (2006), 221-245. MR 2254643 (2007m:22009)
  • [BS] S. Böcherer and R. Schulze-Pillot, Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J. 121(1991), 35-96. MR 1096467 (92f:11066)
  • [C] W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $ G$, Canad. J. Math. 41 (1989), 385-438. MR 1013462 (90j:22013)
  • [Co] J. Cogdell, Lectures on $ L$-functions, converse theorems, and functoriality of $ \operatorname{GL}(n)$, in Lectures on Automorphic $ L$-functions, Fields Institute Monographs, AMS (2004), 5-100. MR 2071506
  • [F1] Y. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. France 116 (1988), 295-313. MR 984899 (89m:11049)
  • [F2] Y. Flicker, On zeros of the twisted tensor $ L$-function, Math. Ann. 297 (1993), 199-219. MR 1241802 (95c:11065)
  • [FZ] Y. Flicker and D. Zinoviev, On poles of twisted tensor $ L$-functions, Proceedings of the Japan Academy 71 (1995), 114-116. MR 1344660 (96f:11075)
  • [HK] M. Harris and S. S. Kudla, Arithmetic automorphic forms for the nonholomorphic discrete series of $ \operatorname{GSp}(2)$, Duke Math. J. 66 (1992), 59-121. MR 1159432 (93e:22023)
  • [HST] M. Harris, D. Soudry and R. Taylor, l-adic representations associated to modular forms over imaginary quadratic fields I: Lifting to $ \operatorname{GSp}_4(\mathbb{Q})$, Invent. Math. 112 (1993), 377-411. MR 1213108 (94d:11035)
  • [HPS] R. Howe and I. I. Piatetski-Shapiro, Some examples of automorphic forms on $ \operatorname{Sp}_4$, Duke Math. J. 50 (1983), 55-106. MR 700131 (84m:10019)
  • [I] T. Ikeda, On the location of the triple $ L$-functions, Compositio Math. 83 (1992), 187-237. MR 1174424 (94b:11042)
  • [J1] H. Jacquet, Principal $ L$-functions of the linear group, in Automorphic Forms, Representations, $ L$-functions, Proc. Symposia Pure Math. vol. XXXIII - Part 2, American Mathematical Society, Providence (1979), 63-86. MR 546609 (81f:22029)
  • [J2] H. Jacquet, : Private communication.
  • [KS] H. Kim and F. Shahidi, Cuspidality of symmetric powers with applications, Duke Math. J. 112 (2002), 177-197 MR 1890650 (2003a:11057)
  • [K] A. Knapp, Representation Theory of Semisimple Groups. An overview based on examples, Princeton University Press, Princeton, NJ, (2001). MR 1880691 (2002k:22011)
  • [Kd1] S. Kudla, On the local theta correspondence, Invent. Math. 83 (1986), 229-255. MR 818351 (87e:22037)
  • [Kd2] S. Kudla, Notes on the Local Theta Correspondence, unpublished notes, available online.
  • [KR1] S. Kudla and S. Rallis, Poles of Eisenstein series and $ L$-functions, Festschrit in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II, Weizmann, Jerusalem, (1990), 81-110. MR 1159110 (94e:11054)
  • [KR2] S. Kudla and S. Rallis, A regularized Siegel-Weil formula: The first term identity, Ann. of Math. (2) 140 (1994), 1-80. MR 1289491 (95f:11036)
  • [M] D. A. Marcus, Number Fields, Springer, (1977). MR 0457396 (56:15601)
  • [P] A. Paul, On the Howe correspondence for symplectic-orthogonal dual pairs, J. Functional Analysis, 228 (2005), 270-310. MR 2175409 (2006g:20076)
  • [PSR] I. I. Piatetski-Shapiro and S. Rallis, $ L$-functions for classical groups, in Lecture Notes in Math. 1254, Springer-Verlag, New York, pp. 1-52.
  • [PSP] D. Prasad and R. Schulze-Pillot, Generalized form of a conjecture of Jacquet and a local consequence, J. Reine Angew. Math. 616 (2008), 219-236. MR 2369492
  • [R1] B. Roberts, The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285-317. MR 1394579 (98a:22007)
  • [R2] B. Roberts, Tempered representations and the theta correspondence, Canad. J. Math. 50 (1998), 1105-1108. MR 1650930 (99j:11054)
  • [R3] B. Roberts, The non-Archimedean theta correspondence for $ \operatorname{GSp}(2)$ and $ \operatorname{GO}(4)$, Trans. AMS 351 (1999), 781-811. MR 1458334 (99d:11056)
  • [R4] B. Roberts, Nonvanishing of global theta lifts from orthogonal groups, J. Ramanujan Math. Soc. 14 (1999), 131-194. MR 1727710 (2000j:11076)
  • [R5] B. Roberts, Global $ L$-packets for $ \operatorname{GSp}(2)$ and theta lifts, Documenta Math. 6 (2001), 247-314. MR 1871665 (2003a:11059)

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Additional Information

Shuichiro Takeda
Affiliation: Department of Mathematics, University of Pennsylvania, 209 South 33rd St., Philadelphia, Pennsylvania 19104-6395
Address at time of publication: Department of Mathematics, Purdue University, 150 N. University, West Lafayette, Indiana 47907

Keywords: Automorphic representation, theta correspondence, theta lifting
Received by editor(s): July 31, 2006
Received by editor(s) in revised form: January 22, 2008
Published electronically: April 10, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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