On gamma factors of Rankin--Selberg integrals for ${\mathrm{U}}_{2\ell }\times {\mathrm{Res}}_{E/F}{\mathrm{GL}}_n$

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 ${\mathrm{U}}_{2\ell }(F)$ and ${\mathrm{GL}}_n(E)$ 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 $(\pi ,{\tau }_1\times {\tau }_2)$ of ${\mathrm{GL}}_{2\ell }(F)$ and ${\mathrm{GL}}_n(F)\times {\mathrm{GL}}_n(F)$. As a corollary, we prove that the gamma factors arising from Langlands--Shahidi method and our gamma factors coincide.

revised extensively following the suggestions by the referee, in particular added a proof of Corollary 1.1. Accepted for publication in J. Number Theory