TEST VECTORS FOR LOCAL CUSPIDAL RANKIN–SELBERG INTEGRALS

Abstract

Let $\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2}$be a pair of cuspidal complex, or $\ell$-adic, representations of the general linear group of rank $n$over a nonarchimedean local field $F$of residual characteristic $p$, different to $\ell$. Whenever the local Rankin–Selberg $L$-factor $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$is nontrivial, we exhibit explicit test vectors in the Whittaker models of $\unicode[STIX]{x1D70B}_{1}$and $\unicode[STIX]{x1D70B}_{2}$such that the local Rankin–Selberg integral associated to these vectors and to the characteristic function of $\mathfrak{o}_{F}^{n}$is equal to $L(X,\unicode[STIX]{x1D70B}_{1},\unicode[STIX]{x1D70B}_{2})$. As an application we prove that the $L$-factor of a pair of banal $\ell$-modular cuspidal representations is the reduction modulo $\ell$of the $L$-factor of any pair of $\ell$-adic lifts.