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Inventiones mathematicae

, Volume 196, Issue 3, pp 651–732 | Cite as

L-functions and theta correspondence for classical groups

  • Shunsuke Yamana
Article

Abstract

Using the doubling method of Piatetski-Shapiro and Rallis, we develop a theory of local factors of representations of classical groups and apply it to give a necessary and sufficient condition for nonvanishing of global theta liftings in terms of analytic properties of the L-functions and local theta correspondence.

Notes

Acknowledgements

We acknowledge the deep influence of the works of Rallis and his collaborators. We thank Stephen Kudla for suggesting the author to prove Proposition 7.1, Atsushi Ichino for his encouragement throughout this project and Binyong Sun and Chen-Bo Zhu for sending us their preprint [55]. A large portion of Sects. 5 and 10 is an outgrowth of discussion with Wee Teck Gan, to whom we are most thankful. This paper was partly written during my stay at the National University of Singapore. We would like to thank the people in NUS for the warm hospitality. The author is partially supported by JSPS Grant-in-Aid for Research Activity Start-up 24840033. This work is partially supported by the JSPS Institutional Program for Young Researcher Overseas Visits “Promoting international young researchers in mathematics and mathematical sciences led by OCAMI”. The anonymous referee deserves special thanks for a very careful reading that led to substantial improvements and clarifications.

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Faculty of MathematicsKyushu UniversityNishi-kuJapan

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