A multivariate integral representation on $\mathrm{GL}_2 \times \mathrm{GSp}_4$ inspired by the pullback formula

TL;DR

This paper introduces a novel two-variable Rankin-Selberg integral inspired by Garrett's pullback formula, representing specific L-functions for generic cusp forms on GL2×GSp4, and provides new insights into their poles.

Contribution

It presents a new integral representation for L-functions on GL2×GSp4 and verifies a case of Jiang's first-term identity, offering a fresh proof regarding poles of these L-functions.

Findings

Established a two-variable integral representing product of L-functions.

Verified a case of Jiang's first-term identity for Eisenstein series.

Provided a new proof for the pole structure of certain L-functions.

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.