Whittaker functions on ${ m GL}_n$ via theta lifting

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.

💡 Research Summary

The paper introduces a novel third method for obtaining explicit formulas for Whittaker functions on general linear groups over Archimedean local fields, complementing the two classical approaches based on Jacquet integrals and on solving systems of partial differential equations. The authors work with the reductive dual pair (GLₙ, GLₙ₊₁) of type II and exploit the explicit theta correspondence to relate Whittaker functions of a generic representation π of GLₙ(F) to those of its theta lift Π = π × 1 on GLₙ₊₁(F).

The first main result (Theorem 3.2) constructs a non‑zero intertwining map
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