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

Author(s): Shuichiro Takeda
Journal: Trans. Amer. Math. Soc. 361 (2009), 5575-5599.
MSC (2000): Primary 11F27
Posted: April 10, 2009
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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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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
Email: stakeda@math.upenn.edu, stakeda@math.purdue.edu

DOI: 10.1090/S0002-9947-09-04787-4
PII: S 0002-9947(09)04787-4
Keywords: Automorphic representation, theta correspondence, theta lifting
Received by editor(s): July 31, 2006
Received by editor(s) in revised form: January 22, 2008
Posted: April 10, 2009
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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