Test vectors for Rankin-Selberg $L$-functions

Abstract

We study the local zeta integrals attached to a pair of generic representations $(\ddot{I}€,\ddot{I}„)$ of $G{L}_n\times G{L}_m$, $n>m$, over a $p$-adic field. Through a process of unipotent averaging we produce a pair of corresponding Whittaker functions whose zeta integral is non-zero, and we express this integral in terms of the Langlands parameters of $\ddot{I}€$ and $\ddot{I}„$. In many cases, these Whittaker functions also serve as a test vector for the associated Rankin-Selberg (local) $L$-function.

arXiv admin note: text overlap with arXiv:1804.07721