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Poles of the Standard
${\mathcal{L}}$-function of 
$G_{2}$ and the Rallis–Schiffmann Lift
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 71 / Issue 5 / October 2019
- Published online by Cambridge University Press:
- 07 March 2019, pp. 1127-1161
- Print publication:
- October 2019
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- Article
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- You have access
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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.
