Extension of Whittaker functions and test vectors

Abstract

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 CPS and Jacquet--Piatetski-Shapiro--Shalika 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.