TEST VECTORS FOR LOCAL CUSPIDAL RANKIN–SELBERG INTEGRALS

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