A multivariate integral representation on 𝐺𝐿₂×𝐺𝑆𝑝₄ inspired by the pullback formula

Pollack, Aaron; Shah, Shrenik

doi:10.1090/tran/7463

A multivariate integral representation on $\mathrm {GL}_2 \times \mathrm {GSp}_4$ inspired by the pullback formula
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by Aaron Pollack and Shrenik Shah PDF

Trans. Amer. Math. Soc. 371 (2019), 5591-5630 Request permission

Abstract:

We give a two variable Rankin–Selberg integral inspired by consideration of Garrett’s pullback formula. For a globally generic cusp form on $\mathrm {GL}_2\times \mathrm {GSp}_4$, the integral represents the product of the $\mathrm {Std}\times \mathrm {Spin}$ and $\mathbf {1} \times \mathrm {Std}$ $L$-functions. We prove a result concerning an Archimedean principal series representation in order to verify a case of Jiang’s first-term identity relating certain non-Siegel Eisenstein series on symplectic groups. Using it, we obtain a new proof of a known result concerning possible poles of these $L$-functions.

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