Some localglobal nonvanishing results for theta lifts from orthogonal groups
Author:
Shuichiro Takeda
Journal:
Trans. Amer. Math. Soc. 361 (2009), 55755599
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 nonvanishing of a global theta lift from to in terms of nonvanishing 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 nonvanishing condition of a global theta lift from to in our improved setting. Also we consider nonvanishing conditions of a global theta lift from to 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), 142. MR 1346217 (96h:22003)
 [AP]
 J. Adler and D. Prasad, On certain multiplicity one theorems, Israel J. Math. 153 (2006), 221245. MR 2254643 (2007m:22009)
 [BS]
 S. Böcherer and R. SchulzePillot, Siegel modular forms and theta series attached to quaternion algebras, Nagoya Math. J. 121(1991), 3596. MR 1096467 (92f:11066)
 [C]
 W. Casselman, Canonical extensions of HarishChandra modules to representations of , Canad. J. Math. 41 (1989), 385438. MR 1013462 (90j:22013)
 [Co]
 J. Cogdell, Lectures on functions, converse theorems, and functoriality of , in Lectures on Automorphic functions, Fields Institute Monographs, AMS (2004), 5100. MR 2071506
 [F1]
 Y. Flicker, Twisted tensors and Euler products, Bull. Soc. Math. France 116 (1988), 295313. MR 984899 (89m:11049)
 [F2]
 Y. Flicker, On zeros of the twisted tensor function, Math. Ann. 297 (1993), 199219. MR 1241802 (95c:11065)
 [FZ]
 Y. Flicker and D. Zinoviev, On poles of twisted tensor functions, Proceedings of the Japan Academy 71 (1995), 114116. MR 1344660 (96f:11075)
 [HK]
 M. Harris and S. S. Kudla, Arithmetic automorphic forms for the nonholomorphic discrete series of , Duke Math. J. 66 (1992), 59121. MR 1159432 (93e:22023)
 [HST]
 M. Harris, D. Soudry and R. Taylor, ladic representations associated to modular forms over imaginary quadratic fields I: Lifting to , Invent. Math. 112 (1993), 377411. MR 1213108 (94d:11035)
 [HPS]
 R. Howe and I. I. PiatetskiShapiro, Some examples of automorphic forms on , Duke Math. J. 50 (1983), 55106. MR 700131 (84m:10019)
 [I]
 T. Ikeda, On the location of the triple functions, Compositio Math. 83 (1992), 187237. MR 1174424 (94b:11042)
 [J1]
 H. Jacquet, Principal functions of the linear group, in Automorphic Forms, Representations, functions, Proc. Symposia Pure Math. vol. XXXIII  Part 2, American Mathematical Society, Providence (1979), 6386. 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), 177197 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), 229255. 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 functions, Festschrit in honor of I. I. PiatetskiShapiro on the occasion of his sixtieth birthday, Part II, Weizmann, Jerusalem, (1990), 81110. MR 1159110 (94e:11054)
 [KR2]
 S. Kudla and S. Rallis, A regularized SiegelWeil formula: The first term identity, Ann. of Math. (2) 140 (1994), 180. MR 1289491 (95f:11036)
 [M]
 D. A. Marcus, Number Fields, Springer, (1977). MR 0457396 (56:15601)
 [P]
 A. Paul, On the Howe correspondence for symplecticorthogonal dual pairs, J. Functional Analysis, 228 (2005), 270310. MR 2175409 (2006g:20076)
 [PSR]
 I. I. PiatetskiShapiro and S. Rallis, functions for classical groups, in Lecture Notes in Math. 1254, SpringerVerlag, New York, pp. 152.
 [PSP]
 D. Prasad and R. SchulzePillot, Generalized form of a conjecture of Jacquet and a local consequence, J. Reine Angew. Math. 616 (2008), 219236. MR 2369492
 [R1]
 B. Roberts, The theta correspondence for similitudes, Israel J. Math. 94 (1996), 285317. MR 1394579 (98a:22007)
 [R2]
 B. Roberts, Tempered representations and the theta correspondence, Canad. J. Math. 50 (1998), 11051108. MR 1650930 (99j:11054)
 [R3]
 B. Roberts, The nonArchimedean theta correspondence for and , Trans. AMS 351 (1999), 781811. MR 1458334 (99d:11056)
 [R4]
 B. Roberts, Nonvanishing of global theta lifts from orthogonal groups, J. Ramanujan Math. Soc. 14 (1999), 131194. MR 1727710 (2000j:11076)
 [R5]
 B. Roberts, Global packets for and theta lifts, Documenta Math. 6 (2001), 247314. 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 191046395
Address at time of publication:
Department of Mathematics, Purdue University, 150 N. University, West Lafayette, Indiana 47907
Email:
stakeda@math.upenn.edu, stakeda@math.purdue.edu
DOI:
http://dx.doi.org/10.1090/S0002994709047874
PII:
S 00029947(09)047874
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.
