Whittaker functions on ${{\rm GL}}_n$ via theta lifting

Abstract

In the literature, two main approaches have been used to establish explicit formulas or propagation formulas for Whittaker functions over Archimedean local fields: one based on Jacquet integrals, and the other on the analysis of systems of partial differential equations. In this paper, we introduce a third approach via explicit theta correspondence. As an example, we derive new cases of explicit formulas for Whittaker functions on ${{\rm GL}}_n(\mathbb{C})$ and compute the associated Asai local zeta integrals.