Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture

Abstract

In this article, we construct a family of integrals which represent the product of Rankin-Selberg $L$-functions of ${\mathrm{GL}}_l\times {\mathrm{GL}}_m$ and of ${\mathrm{GL}}_l\times {\mathrm{GL}}_n$ when $m+n<l$. When $n=0$, these integrals are those defined by Jacquet--Piatetski-Shapiro--Shalika up to a shift. In this sense, these new integrals generalize Jacquet--Piatetski-Shapiro--Shalika's Rankin-Selberg convolution integrals. We study basic properties of these integrals. In particular, we define local gamma factors using this new family of integrals. As an application, we obtain a new proof of Jacquet's local converse conjecture using these new integrals.