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

Gurevich, Nadya ; Segal, Avner

Canadian journal of mathematics, Tome 71 (2019) no. 5, pp. 1127-1161

Résumé

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.

Détail

DOI : 10.4153/CJM-2018-019-7

Mots-clés : automorphic representation, exceptional theta-lift, Siegel-Weil identity

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