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Canadian Journal of Mathematics Canadian Journal of Mathematics

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Poles of the Standard ${\mathcal{L}}$ -function of $G_{2}$ and the Rallis–Schiffmann Lift

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  07 March 2019

Nadya Gurevich  and
Avner Segal
Nadya Gurevich
Affiliation:
School of Mathematics, Ben Gurion University of the Negev, POB 653, Be’er Sheva 84105, Israel Email: ngur@math.bgu.ac.il
Avner Segal
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel University of British Columbia, Vancouver BC V6T 1Z2 Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel Email: segalavner@gmail.com
Corresponding
E-mail address:
ngur@math.bgu.ac.il
segalavner@gmail.com
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Abstract

We characterize the cuspidal representations of $G_{2}$ whose standard ${\mathcal{L}}$ -function admits a pole at $s=2$ as the image of the Rallis–Schiffmann lift for the commuting pair ( $\widetilde{\text{SL}}_{2}$ , $G_{2}$ ) in $\widetilde{\text{Sp}}_{14}$ . The image consists of non-tempered representations. The main tool is the recent construction, by the second author, of a family of Rankin–Selberg integrals representing the standard ${\mathcal{L}}$ -function.

Keywords

automorphic representation exceptional theta-lift Siegel-Weil identity

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 11F27: Theta series; Weil representation; theta correspondences 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Type
Article
Information
Canadian Journal of Mathematics , Volume 71 , Issue 5 , October 2019 , pp. 1127 - 1161
Copyright
© Canadian Mathematical Society 2019 

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Footnotes

The authors were partially supported by grants 1691/10 and 259/14 from the Israel Science Foundation.

References

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