Extension of Whittaker functions and test vectors

We show that certain products of Whittaker functions and Schwartz functions on a general linear group extend to Whittaker functions on a larger general linear group. This generalizes results of Cogdell–Piatetski-Shapiro \cite{CPS} and Jacquet–Piatetski-Shapiro–Shalika \cite{JPSS83}. As a consequence, we prove that the Rankin–Selberg $L$-factor of the product of a discrete series representation and the Zelevinsky dual of a discrete series representation is given by a single Rankin–Selberg integral.