PRODUCT OF RANKIN-SELBERG CONVOLUTIONS AND A NEW PROOF OF JACQUET’S LOCAL CONVERSE CONJECTURE

Abstract

This article constructs a family of integrals representing the product of Rankin–Selberg L -functions for

and

GL l × GL n ${\mathrm {GL}}_l \times {\mathrm {GL}}_n$ upper G upper L Subscript l Baseline times upper G upper L Subscript n

in the regime

m + n < l $m+n<l$ m plus n less than l

. When

n = 0 $n=0$ n equals 0

, these integrals reduce (up to a shift) to the classical Rankin-Selberg convolution integrals of Jacquet–Piatetski-Shapiro–Shalika (JPSS), thereby generalizing the JPSS integrals. We establish their basic properties and use them to define local gamma factors. As an application, we provide a new proof of Jacquet’s local converse conjecture.