On gamma factors of Rankin–Selberg integrals for U2ℓ × ResE/FGLn

Abstract

In this paper, we prove the fundamental properties of gamma factors defined by Rankin-Selberg integrals of Shimura type for pairs of generic representations $(\pi ,\tau )$ of $U_{2\ell }\left(F\right)$ and ${\mathrm{GL}}_n\left(E\right)$ for a local field F of characteristic zero and a quadratic extension E of F. We also prove similar results for pairs of generic representations $(\pi ,{\tau }_1\otimes {\tau }_2)$ of ${\mathrm{GL}}_{2\ell }\left(F\right)$ and ${\mathrm{GL}}_n\left(F\right)\times {\mathrm{GL}}_n\left(F\right)$. As a corollary, we prove that the gamma factors arising from Langlands–Shahidi method and our gamma factors coincide.