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Poles of the Standard $\mathcal{L}$-function of $G_2$ and the Rallis-Schiffmann lift

Published:2019-01-21

School of Mathematics, Ben Gurion University of the Negev, POB 653, Be'er Sheva 84105, Israel
• Avner Segal,
School of Mathematics, Ben Gurion University of the Negev, POB 653, Be'er Sheva 84105, Israel
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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 $(\widetilde{SL_2}, G_2)$ in $\widetilde{Sp_{14}}$. 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 Classifications: 11F70 - Representation-theoretic methods; automorphic representations over local and global fields 11F27 - Theta series; Weil representation; theta correspondences 11F66 - Langlands $L$-functions; one variable Dirichlet series and functional equations

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