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

TL;DR

This paper introduces a novel approach using explicit theta correspondence to derive formulas for Whittaker functions on GL(n) over complex fields, expanding the methods beyond traditional integral and PDE analyses.

Contribution

It presents a new method via theta lifting for explicit formulas of Whittaker functions, specifically on GL(n) over complex numbers, and computes related Asai local zeta integrals.

Findings

Derived new explicit formulas for Whittaker functions on GL(n,C)

Computed Asai local zeta integrals associated with these functions

Demonstrated the effectiveness of theta correspondence in this context

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.