Extension of Whittaker functions and test vectors

TL;DR

This paper extends the theory of Whittaker functions on general linear groups, showing that certain products can be extended to larger groups, and proves a key property of Rankin--Selberg L-factors for discrete series representations.

Contribution

It generalizes previous results on Whittaker functions and establishes that the Rankin--Selberg L-factor for specific representation products equals a single integral.

Findings

Products of Whittaker and Schwartz functions extend to larger groups.

Rankin--Selberg L-factors for discrete series are given by a single integral.

Generalizes prior results of Cogdell--Piatetski-Shapiro and Jacquet--Piatetski-Shapiro--Shalika.

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 \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.